Quantum Heisenberg models and their probabilistic representations
EEntropy & the Quantum II, Contemporary Mathematics 552, 177–224 (2011)
Quantum Heisenberg models and their probabilisticrepresentations
Christina Goldschmidt, Daniel Ueltschi, and Peter Windridge
Abstract.
These notes give a mathematical introduction to two seeminglyunrelated topics: (i) quantum spin systems and their cycle and loop representa-tions, due to T´oth and Aizenman-Nachtergaele; (ii) coagulation-fragmentationstochastic processes. These topics are nonetheless related, as we argue thatthe lengths of cycles and loops effectively perform a coagulation-fragmentationprocess. This suggests that their joint distribution is Poisson-Dirichlet. Theseideas are far from being proved, but they are backed by several rigorous results,notably of Dyson-Lieb-Simon and Schramm.
Contents
1. Introduction 1781.1. Guide to notation 1792. Hilbert space, spin operators, Heisenberg Hamiltonian 1802.1. Graphs and Hilbert space 1802.2. Spin operators 1812.3. Hamiltonians and magnetization 1822.4. Gibbs states and free energy 1822.5. Symmetries 1843. Stochastic representations 1843.1. Poisson edge process, cycles and loops 1843.2. Duhamel expansion 1863.3. T´oth’s representation of the ferromagnet 1873.4. Aizenman-Nachtergaele’s representation of the antiferromagnet 1894. Thermodynamic limit and phase transitions 1914.1. Thermodynamic limit 1924.2. Ferromagnetic phase transition 1934.3. Antiferromagnetic phase transition 195
Mathematics Subject Classification.
Key words and phrases.
Spin systems, quantum Heisenberg model, probabilistic representa-tions, Poisson-Dirichlet distribution, split-merge process.Work partially supported by EPSRC grant EP/G056390/1.c (cid:13) c (cid:13) a r X i v : . [ m a t h - ph ] J u l
78 C. GOLDSCHMIDT, D. UELTSCHI, AND P. WINDRIDGE
1. Introduction
We review cycle and loop models that arise from quantum Heisenberg spinsystems. The loops and cycles are geometric objects defined on graphs. The maingoal is to understand properties such as their length in large graphs.The cycle model was introduced by T´oth as a probabilistic representation ofthe Heisenberg ferromagnet [ ], while the loop model is due to Aizenman andNachtergaele and is related to the Heisenberg antiferromagnet [ ]. Both models arebuilt on the random stirring process of Harris [ ] and have an additional geometricweight of the form ϑ or ϑ with parameter ϑ = 2. Recently, Schrammstudied the cycle model on the complete graph and with ϑ = 1 (that is, withoutthis factor) [ ]. He showed in particular that cycle lengths are generated by asplit-merge process (or “coagulation-fragmentation” process), and that the cyclelengths have Poisson-Dirichlet distribution with parameter 1.The graphs of physical relevance are regular lattices such as Z d (or large fi-nite boxes in Z d ), and the factor 2 needs to be present. What should weexpect in this case? A few hints come from the models of spatial random per-mutations, which also involve one-dimensional objects living in higher dimensionalspaces. The average length of the longest cycle in lattice permutations was com-puted numerically in [ ]. In retrospect, this suggests that the cycle lengths havethe Poisson-Dirichlet distribution. In the “annealed” model where positions areaveraged, this was proved in [ ]; the mechanisms at work there (i.e., Bose-Einsteincondensation and non-spatial random permutations with Ewens distribution), how-ever, seem very specific.We study the cycle and loop models in Z d with the help of a stochastic processwhose invariant measure is identical to the original measure with weight ϑ or ϑ , and which leads to an effective split-merge process for the cycle (or loop) EISENBERG MODELS AND THEIR PROBABILISTIC REPRESENTATIONS 179 lengths. The rates at which the splits and the merges take place depends on ϑ . Thisallows us to identify the invariant measure, which turns out to be Poisson-Dirichletwith parameter ϑ . While we cannot make these ideas mathematically rigorous,they are compatible with existing results.As mentioned above, cycle and loop models are closely related to Heisenbergmodels. In particular, the cycle and loop geometry is reflected in some importantquantum observables. These observables have been the focus of intense study bymathematical and condensed matter physicists, who have used imagination andclever observations to obtain remarkable results in the last few decades. Mostrelevant to us is the theorem of Mermin and Wagner about the absence of magneticorder in one and two dimensions [ ], and the theorem of Dyson, Lieb, and Simon,about the existence of magnetic order in the antiferromagnetic model in dimensions3 and more [ ]. We review these results and explain their implications for cycleand loop models.Many a mathematician is disoriented when wandering in the realm of quantumspin systems. The landscape of 2 × ], p. 314, betweenanalysts’ language and probabilists’ “dialect”). In these notes, we have attemptedto introduce these different notions in a self-contained fashion. The following objects play a central rˆole.Λ = ( V , E ) A finite graph with undirected edges. S ( j ) x , (cid:126)S x Spin operators ( § (cid:104)·(cid:105) Λ ,β,h Gibbs state ( § Z Λ , F Λ Partition function and free energy ( § ρ Λ ,β ( dω ) Probability measure for Poisson point processes on [0 , β ] ( β >
0) attached to each edge of Λ (defined in § C ( ω ) , L ( ω ) Cycle and loop configurations constructed from edges in ω ( § γ Cycle in C ( ω ) or loop in L ( ω ). ϑ > n = ( V n , E n ) Box { , . . . , n } d in Z d with nearest-neigbor edges ( § m ∗ th , m ∗ res , m ∗ sp Various definitions of the magnetization ( § σ ( β ) Antiferromagnetic long-range order ( § η ∞ , η macro Fractions of vertices in infinite or macroscopic cycles/loops( § Countable partitions of [0 ,
1] with parts in decreasing order,i.e. { p ≥ p ≥ . . . ≥ (cid:80) i p i = 1 } .PD θ Poisson-Dirichlet distribution with parameter θ on ∆ ( § X t , t ≥
0) Stochastic process with invariant measure given by our cycleand loop models ( §
80 C. GOLDSCHMIDT, D. UELTSCHI, AND P. WINDRIDGE
2. Hilbert space, spin operators, Heisenberg Hamiltonian
We review the setting for quantum lattice spin systems described by Heisenbergmodels. Spin systems are relevant for the study of electronic properties of condensedmatter. Atoms form a regular lattice and they host localized electrons, whichare characterized only by their spin. Interactions are restricted to neighboringspins. One is interested in equilibrium properties of large systems. There aretwo closely related quantum Heisenberg models, which describe ferromagnets andantiferromagnets, respectively. The material is standard and the interested readeris encouraged to look in the references [
44, 46, 39, 19 ] for further information.
Let Λ = ( V , E ) be a graph, where V is afinite set of vertices and E is the set of “edges”, i.e. unordered pairs in V × V . Froma physical perspective, relevant graphs are regular graphs such as Z d (or a finite boxin Z d ) with nearest-neighbor edges, but it is mathematically advantageous to allowfor more general graphs. We restrict ourselves to spin- systems, mainly becausethe stochastic representations only work in this case.To each site x ∈ V is associated a 2-dimensional Hilbert space H x = C . It isconvenient to use Dirac’s “bra”, (cid:104)·| , and “ket”, |·(cid:105) , notation, in which we identify | (cid:105) = (cid:18) (cid:19) , | − (cid:105) = (cid:18) (cid:19) . (2.1) The notation (cid:104) f | g (cid:105) means the inner product; we use the convention that it is linearin the second variable (and antilinear in the first). Occasionally, we also write (cid:104) f | A | g (cid:105) for (cid:104) f | Ag (cid:105) . The Hilbert space of a quantum spin system on Λ is the tensorproduct H ( V ) = (cid:79) x ∈ Λ H x , (2.2) which is the 2 |V| dimensional space spanned by elements of the form ⊗ x ∈V f x with f x ∈ H x . The inner product between two such vectors is defined by (cid:68) ⊗ x ∈V f x (cid:12)(cid:12)(cid:12) ⊗ x ∈V g x (cid:69) H ( V ) = (cid:89) x ∈ Λ (cid:104) f x | g x (cid:105) H x . (2.3) The inner product above extends by (anti)linearity to the other vectors, which areall linear combinations of vectors of the form ⊗ x ∈V f x .The basis (2.1) of C has a natural extension in H ( V ) ; namely, given s ( V ) =( s x ) x ∈V with s x = ± , let | s ( V ) (cid:105) = (cid:79) x ∈ Λ | s x (cid:105) . (2.4) These elements are orthonormal, i.e. (cid:104) s ( V ) | ˜ s ( V ) (cid:105) = (cid:89) x ∈V (cid:104) s x | ˜ s x (cid:105) = (cid:89) x ∈V δ s x , ˜ s x , (2.5) where δ is Kronecker’s symbol, δ ab = 1 if a = b , 0 otherwise. EISENBERG MODELS AND THEIR PROBABILISTIC REPRESENTATIONS 181
In the quantum world, physically relevant quantitiesare called observables and they are represented by self-adjoint operators. The op-erators for the observable properties of our spin- particles are called the Paulimatrices, defined by S (1) = (cid:18) (cid:19) , S (2) = (cid:18) − ii 0 (cid:19) , S (3) = (cid:18) − (cid:19) . (2.6) We interpret S ( i ) as the spin component in the i th direction. The matrices areclearly Hermitian and satisfy the relations[ S (1) , S (2) ] = i S (3) , [ S (2) , S (3) ] = i S (1) , [ S (3) , S (1) ] = i S (2) . (2.7) These operators have natural extensions as spin operators in H ( V ) . Let x ∈ V ,and write H ( V ) = H x ⊗ H ( V\{ x } ) . We define operators S ( i ) x indexed by x ∈ V by S ( i ) x = S ( i ) ⊗ Id V\{ x } . (2.8) The commutation relations (2.7) extend to the operators S ( i ) x , namely[ S (1) x , S (2) y ] = i δ xy S (3) x , (2.9) and all other relations obtained by cyclic permutations of (123). Indeed, it is nothard to check that the matrix elements (cid:104) s ( V ) |·| ˜ s ( V ) (cid:105) of both sides are identical for all s ( V ) ∈ {− , } V . It is customary to introduce the notation (cid:126)S x = ( S (1) x , S (2) x , S (3) x ),and (cid:126)S x · (cid:126)S y = S (1) x S (1) y + S (2) x S (2) y + S (3) x S (3) y . (2.10) Note that operators of the form S ( i ) x S ( j ) y , with x (cid:54) = y , act in H ( V ) = H x ⊗ H y ⊗H ( V\{ x,y } ) as follows S ( i ) x S ( j ) y = S ( i ) ⊗ S ( j ) ⊗ Id V\{ x,y } . (2.11) In the case x = y , and using ( S ( i ) x ) = Id V , we get (cid:126)S x = ( S (1) x ) + ( S (2) x ) + ( S (3) x ) = Id V . (2.12) Lemma 2.1.
Consider (cid:126)S x · (cid:126)S y in H x ⊗ H y . It is self-adjoint, and its eigenvaluesand eigenvectors are as follows: • − is an eigenvalue with multiplicity 1; the eigenvector is √ ( | , − (cid:105) −| − , (cid:105) ) . • is an eigenvalue with multiplicity 3; three orthonormal eigenvectors are | , (cid:105) , | − , − (cid:105) , √ (cid:0) | , − (cid:105) + | − , (cid:105) (cid:1) . The eigenvector corresponding to − is called a “singlet state” by physicists,while the eigenvectors for are called “triplet states”. Proof.
We have for all a, b = ± , S (1) x S (1) y | a, b (cid:105) = | − a, − b (cid:105) ,S (2) x S (2) y | a, b (cid:105) = − ab | − a, − b (cid:105) ,S (3) x S (3) y | a, b (cid:105) = ab | a, b (cid:105) . (2.13) The lemma then follows from straightforward linear algebra. (cid:3)
82 C. GOLDSCHMIDT, D. UELTSCHI, AND P. WINDRIDGE
We can now introduce the Heisen-berg Hamiltonians, which are self-adjoint operators in H ( V ) . H ferroΛ ,h = − (cid:88) { x,y }∈E (cid:126)S x · (cid:126)S y − h (cid:88) x ∈V S (3) x ,H antiΛ ,h = + (cid:88) { x,y }∈E (cid:126)S x · (cid:126)S y − h (cid:88) x ∈V S (3) x . (2.14) Let us briefly discuss the physical motivation behind these operators. One is in-terested in describing a condensed matter system where atoms are arranged on aregular lattice. Each atom hosts exactly one relevant electron. Each electron stayson its atom and its spin is described by a vector in the Hilbert space C . A systemof two spins is described by a vector in C ⊗ C . The singlet and triplet statesof Lemma 2.1 are invariant under rotation of the spins and they form a basis. Inabsence of external magnetic field, the energy operator should be diagonal withrespect to these states, and there should be one eigenvalue for the singlet, and oneother eigenvalue for the triplets. Up to constants, it should then be ± (cid:126)S x · (cid:126)S y . Itis natural to define the total energy as the sum of nearest-neighbor interactions.Taking into account the contribution of the external magnetic field, which can bejustified along similar lines, we get the Hamiltonians of (2.14).Next, let M Λ be the operator that represents the magnetization in the 3rddirection. M (3)Λ = (cid:88) x ∈V S (3) x . (2.15) Lemma 2.2.
The Hamiltonian and magnetization operators commute, i.e., [ H Λ ,h , M Λ ] = 0 . Proof.
This follows from the commutation relations (2.9). Namely, using thefact that S ( i ) x and S (3) y commute for x (cid:54) = y ,[ H Λ ,h , M Λ ] = (cid:88) { x,y }∈E ,z ∈V [ (cid:126)S x · (cid:126)S y , S (3) z ]= (cid:88) { x,y }∈E (cid:16) [ S (1) x S (1) y , S (3) x ] + [ S (1) x S (1) y , S (3) y ] + [ S (2) x S (2) y , S (3) x ] + [ S (2) x S (2) y , S (3) y ] (cid:17) . (2.16) The first commutator is[ S (1) x S (1) y , S (3) x ] = [ S (1) x , S (3) x ] S (1) y = − i S (2) x S (1) y , (2.17) and the others are similar. We get[ H Λ ,h , M Λ ] = i (cid:88) { x,y }∈E (cid:16) − S (2) x S (1) y − S (1) x S (2) y + S (1) x S (2) y + S (2) x S (1) y (cid:17) = 0 . (2.18) (cid:3) The equilibrium states of quantumstatistical mechanics are given by
Gibbs states (cid:104)·(cid:105) Λ ,β,h . These are nonnegativelinear functionals on the space of operators in H ( V ) of the form (cid:104) A (cid:105) Λ ,β,h = 1 Z Λ ( β, h ) Tr A e − βH Λ ,h , (2.19) EISENBERG MODELS AND THEIR PROBABILISTIC REPRESENTATIONS 183 where the normalization Z Λ ( β, h ) = Tr e − βH Λ ,h (2.20) is called the partition function . Here, Tr denotes the usual matrix trace.There are deep reasons why the Gibbs states describe equilibrium states butwe will not dwell on them here. We now introduce the free energy F Λ ( β, h ). Itsphysical motivation is that it provides a connection to thermodynamics. It is a kindof generating function and it is therefore mathematically useful. The definition ofthe free energy in our case is F Λ ( β, h ) = − β log Z Λ ( β, h ) . (2.21) Lemma 2.3.
The function βF Λ ( β, h ) is concave in ( β, βh ) . Proof.
We will rather check that − βF Λ is convex, which is the case if thematrix (cid:32) ∂ βF Λ ∂β ∂ βF Λ ∂β∂ ( βh ) ∂ βF Λ ∂β∂ ( βh ) ∂ βF Λ ∂ ( βh ) (cid:33) is positive definite. Let us write (cid:104)·(cid:105) instead of (cid:104)·(cid:105) Λ ,β,h . We have ∂ ∂β βF Λ ( β, h ) = − (cid:10) ( H Λ , − (cid:104) H Λ , (cid:105) ) (cid:11) ,∂ ∂ ( βh ) βF Λ ( β, h ) = − (cid:10) ( M Λ − (cid:104) M Λ (cid:105) ) (cid:11) ,∂ ∂β∂ ( βh ) βF Λ ( β, h ) = (cid:10) ( H Λ , − (cid:104) H Λ , (cid:105) )( M Λ − (cid:104) M Λ (cid:105) ) (cid:11) . (2.22) Then F Λ is convex if (cid:10) ( H Λ , − (cid:104) H Λ , (cid:105) )( M Λ − (cid:104) M Λ (cid:105) ) (cid:11) ≤ (cid:10) ( H Λ , − (cid:104) H Λ , (cid:105) ) (cid:11)(cid:10) ( M Λ − (cid:104) M Λ (cid:105) ) (cid:11) . (2.23) It is not hard to check that the map (
A, B ) (cid:55)→ (cid:104) A ∗ B (cid:105) is an inner product on thespace of operators that commute with H Λ ,h . Then |(cid:104) A ∗ B (cid:105)| ≤ (cid:104) A ∗ A (cid:105)(cid:104) B ∗ B (cid:105) (2.24) by the Cauchy-Schwarz inequality and, in particular, this implies (2.23). (cid:3) Concave functions are necessarily continuous. But it is useful to establish that F Λ ( β, h ) is uniformly continuous on compact domains. This property will be usedin Section 4.1 which discusses the existence of infinite volume limits. Lemma 2.4. (cid:12)(cid:12) βF Λ ( β, h ) − β (cid:48) F Λ ( β (cid:48) , h (cid:48) ) (cid:12)(cid:12) ≤ | β − β (cid:48) | ( |E| + | h | |V| ) + β | h − h (cid:48) ||V| . Proof.
We have βF Λ ( β, h ) − β (cid:48) F Λ ( β (cid:48) , h ) = (cid:90) ββ (cid:48) dd s sF Λ ( s, h )d s = (cid:90) ββ (cid:48) (cid:104) H Λ ,h (cid:105) Λ ,s,h d s. (2.25) We can also check that βF Λ ( β, h ) − βF Λ ( β, h (cid:48) ) = (cid:82) hh (cid:48) (cid:104) M Λ (cid:105) Λ ,β,s d s . The result followsfrom |(cid:104) A (cid:105) Λ ,β,h | ≤ (cid:107) A (cid:107) for any operator A , and from (cid:107) (cid:126)S x · (cid:126)S y (cid:107) = (cf Lemma 2.1)and (cid:107) S (3) x (cid:107) = . (cid:3)
84 C. GOLDSCHMIDT, D. UELTSCHI, AND P. WINDRIDGE
In quantum statistical mechanics, a symmetry is repre-sented by a unitary transformation which leaves the Hamiltonian invariant. Itfollows that (finite volume) Gibbs states also possess the symmetry. However,infinite volume states may lose it. This is called symmetry breaking and is a man-ifestation of a phase transition. We only mention the “spin flip” symmetry here,corresponding to the unitary operator U | s ( V ) (cid:105) = | − s ( V ) (cid:105) . (2.26) One can check that U − S ( i ) x S ( i ) y U = S ( i ) x S ( i ) y and U − S (3) x U = − S (3) x . It followsthat U − H Λ ,h U = H Λ , − h . (2.27) This applies to both the ferromagnetic and antiferromagnetic Hamiltonians. Itfollows that F Λ ( β, − h ) = F Λ ( β, h ), and so the free energy is symmetric as a functionof h .
3. Stochastic representations
Stochastic representations of quantum lattice models go back to Ginibre, whoused a Peierls contour argument to prove the occurrence of phase transitions inanisotropic models [ ]. Conlon and Solovej introduced a random walk represen-tation for the ferromagnetic model and used it to get an upper bound on the freeenergy [ ]. A different representation was introduced by T´oth, who improvedthe previous bound [ ]. Further work on quantum models using similar repre-sentations include the quantum Pirogov-Sinai theory [
11, 15 ] and Ising models intransverse magnetic field [
30, 14, 28 ].A major advantage of T´oth’s representation is that spin correlations have natu-ral probabilistic expressions, being given by the probability that two sites belong tothe same cycle (see below for details). A similar representation was introduced byAizenman and Nachtergaele for the antiferromagnetic model, who used it to studyproperties of spin chains [ ]. The random objects are a bit different (loops insteadof cycles), but this representation shares the advantage that spin correlations aregiven by the probability of belonging to the same loop.The representations due to T´oth and Aizenman-Nachtergaele both involve aPoisson process on the edges of the graph. The measure is reweighted by a functionof suitable geometric objects (“cycles” or “loops”). We first describe the two modelsin Section 3.1; we will relate them to the Heisenberg models in Sections 3.3 and3.4. Recall that Λ = ( V , E ) isa finite undirected graph. We attach to each edge a Poisson process on [0 , β ] ofunit intensity (see § ω = (cid:0) ( e , t ) , . . . , ( e k , t k ) (cid:1) . (3.1) Each pair is called a bridge. The number of bridges across each edge, thus, hasa Poisson distribution with mean β , and the total number of bridges is Poissonwith mean β |E| . Conditional on there being k bridges, their times of arrival areuniformly distributed in { < t < t < . . . < t k < β } and the edges are chosenuniformly from E . The corresponding measure is denoted ρ Λ ,β (d ω ). EISENBERG MODELS AND THEIR PROBABILISTIC REPRESENTATIONS 185
To each realization ω there corresponds a configuration of cycles and configura-tion of loops. The mathematical definitions are a bit cumbersome but the geometricideas are simpler and more elegant. The reader is encouraged to look at Figure 1for an illustration. cycles loopsA B A ABBA Figure 1.
Top: an edge Poisson configuration ω on V × [0 , β ] per .Bottom left: its associated cycle configuration. Bottom right: itsassociated loop configuration. We see that |C ( ω ) | = 3 and |L ( ω ) | =5.We consider the cylinder V × [0 , β ] per , where the subscript “per” indicatesthat we consider periodic boundary conditions. A cycle is a closed trajectory onthis space; that is, it is a function γ : [0 , L ] → V × [0 , β ] per such that, if γ ( τ ) =( x ( τ ) , t ( τ )), we have: • γ ( τ ) is piecewise continuous; if it is continuous on the interval I ⊂ [0 , L ],then x ( τ ) is constant and dd τ t ( τ ) = 1 in I . • γ ( τ ) is discontinuous at τ iff the pair ( e, t ) belongs to ω , where t = t ( τ )and e is the edge { x ( τ − ) , x ( τ +) } .We choose L to be the smallest positive number such that γ ( L ) = γ (0). Then L is the length of the cycle; it corresponds to the sum of the vertical legs in Figure1 and is necessarily a multiple of β . Let us make the cycles semi-continuous byassigning the value γ ( τ ) = γ ( τ − ) at the points of discontinuity. We identify cycleswhose support is identical. Then to each ω corresponds a configuration of cycles C ( ω ) whose supports form a partition of the cylinder V × [0 , β ] per . The number ofcycles is |C ( ω ) | .Loops are similar, but we now suppose that the graph is bipartite. The Asublattice possesses and orientation, which is reversed on the B sublattice. We still
86 C. GOLDSCHMIDT, D. UELTSCHI, AND P. WINDRIDGE consider the cylinder
V × [0 , β ] per . A loop is a closed trajectory on this space; thatis, it is a function γ : [0 , L ] → V × [0 , β ] per such that, with γ ( τ ) = ( x ( τ ) , t ( τ )): • γ ( τ ) is piecewise continuous; if it is continuous in interval I ⊂ [0 , L ], then x ( τ ) is constant and, in I ,dd τ t ( τ ) = (cid:40) x ( τ ) belongs to the A sublattice, − x ( τ ) belongs to the B sublattice. (3.2) • γ ( τ ) is discontinuous at τ iff the pair ( e, t ) belongs to ω , where t = t ( τ )and e is the edge { x ( τ − ) , x ( τ +) } .We choose L to be the smallest positive number such that γ ( L ) = γ (0). Then L is the length of the loop; it corresponds to the sum of the vertical legs in Figure1 (as for cycles), but it is not a multiple of β in general (contrary to cycles). Wealso make the loops semi-continuous by assigning the value γ ( τ ) = γ ( τ − ) at thepoints of discontinuity. Identifying loops whose support is identical, to each ω corresponds a configuration of loops L ( ω ) whose supports form a partition of thecylinder V × [0 , β ] per . The number of loops is |L ( ω ) | .As we shall see, the relevant probability measures for the Heisenberg models(with h = 0) are proportional to 2 |C ( ω ) | ρ E ,β (d ω ) and 2 |L ( ω ) | ρ E ,β (d ω ). We first state and prove Duhamel’s formula. Itis a variant of the Trotter product formula that is usually employed to derivestochastic representations.
Proposition 3.1.
Let
A, B be n × n matrices. Then e A + B = e A + (cid:90) e tA B e (1 − t )( A + B ) d t = (cid:88) k ≥ (cid:90) Let F ( s ) be the matrix-valued function F ( s ) = e sA + (cid:90) s e tA B e ( s − t )( A + B ) d t. (3.3) We show that, for all s , e s ( A + B ) = F ( s ) . (3.4) The derivative of F ( s ) is F (cid:48) ( s ) = e sA A + e sA B + (cid:90) s e tA B e ( s − t )( A + B ) ( A + B )d t = F ( s )( A + B ) . (3.5) On the other hand, the derivative of e s ( A + B ) is e s ( A + B ) ( A + B ). The identity (3.4)clearly holds for s = 0 and, since both sides satisfy the same differential equation,they must be equal for all s .We can iterate Duhamel’s formula N times so as to gete A + B = N (cid:88) k =0 (cid:90) Using (cid:107) e tA (cid:107) ≤ e t (cid:107) A (cid:107) , the last term is less than 2 e (cid:107) A (cid:107) + (cid:107) B (cid:107) (cid:107) B (cid:107) N N ! and so it vanishesin the limit N → ∞ . The summand is less than e (cid:107) A (cid:107) (cid:107) B (cid:107) k k ! , so that the sum isabsolutely convergent. (cid:3) Our goal is to perform Duhamel’s expansion on the Gibbs operator e − βH Λ ,h ,where the Hamiltonian is given by a sum of terms indexed by the edges and byvertices. The following corollary applies in this case. Corollary 3.2. Let A and ( h e ) , e ∈ E , be matrices in H ( V ) . Then e β ( A + (cid:80) e ∈E h e ) = (cid:90) d ρ E ,β ( ω ) e t A h e e ( t − t ) A h e . . . h e k e ( β − t k ) A , where ( t , e ) , . . . , ( t k , e k ) are the bridges in ω . Proof. We can expand the right side by summing over the number k of events,then integrating over 0 < t < · · · < t k < β for the times of occurrence, and thensumming over edges e , . . . , e k ∈ E . After the change of variables t (cid:48) i = t i /β , werecognize the second line of Proposition 3.1. (cid:3) It is convenient to intro-duce the operator T x,y which transposes the spins at x and y . In H x ⊗ H y , theoperator acts as follows: T x,y | a, b (cid:105) = | b, a (cid:105) , a, b = ± . (3.7) This rule extends to general vectors by linearity, and it extends to H ( V ) by tensoringit with Id V\{ x,y } . Using Lemma 2.1, it is not hard to check that (cid:126)S x · (cid:126)S y = T x,y − Id { x,y } . (3.8) Recall that C ( ω ) is the set of cycles of ω , and let γ x ∈ C ( ω ) denote the cyclethat intersects ( x, ∈ V × [0 , β ] per . Let L ( γ ) denote the (vertical) length of thecycle γ ; it is always a multiple of β in the theorem below. Theorem 3.3 (T´oth’s representation of the ferromagnet) . The partition function,the average magnetization, and the two-point correlation function have the followingexpressions. Z ferroΛ (2 β, h ) = e − β |E| (cid:90) d ρ E ,β ( ω ) (cid:89) γ ∈C ( ω ) (cid:0) hL ( γ )) (cid:1) , Tr S (3) x e − βH ferroΛ ,h = e − β |E| (cid:90) d ρ E ,β ( ω ) tanh( hL ( γ x )) (cid:89) γ ∈C ( ω ) (cid:0) hL ( γ )) (cid:1) , Tr S (3) x S (3) y e − βH ferroΛ ,h = e − β |E| (cid:90) d ρ E ,β ( ω ) (cid:89) γ ∈C ( ω ) (cid:0) hL ( γ )) (cid:1) × (cid:40) if γ x = γ y , tanh( hL ( γ x )) tanh( hL ( γ y )) if γ x (cid:54) = γ y . 88 C. GOLDSCHMIDT, D. UELTSCHI, AND P. WINDRIDGE 12 121212 12 12 12 1212 121212 12 12 12 12 Figure 2. Each cycle is characterized by a given spin. Proof. The partition function can be expanded using Corollary 3.2 so as toget Z ferroΛ (2 β, h ) = e − β |E| Tr e β (2 hM Λ + (cid:80) e T e ) = e − β |E| (cid:90) d ρ E ,β ( ω ) (cid:88) s ( V ) (cid:104) s ( V ) | e t hM Λ T e . . . T e k e β − t k ) hM Λ | s ( V ) (cid:105) , (3.9) where ( e , t ) , . . . , ( e k , t k ) are the times and edges of ω . Observe that the vectors | s ( V ) (cid:105) are eigenvectors of e tM Λ . It is not hard to see that the matrix element aboveis zero unless each cycle is characterized by a single spin value (see illustration inFigure 2). If the matrix element is not zero, then it is equal to (cid:104) s ( V ) | e t hM Λ T e . . . T e k e β − t k ) hM Λ | s ( V ) (cid:105) = (cid:89) γ ∈C ( ω ) e hL ( γ ) s ( γ ) (3.10) with s ( γ ) the spin of the cycle γ . After summing over s ( γ ) = ± , each cyclecontributes e hL ( γ ) + e − hL ( γ ) = 2 cosh( hL ( γ )), and we obtain the expression forthe partition function.The expression which involves S (3) x is similar, except that the cycle γ x thatcontains x ×{ } contributes e hL ( γ x ) − e − hL ( γ x ) = sinh( hL ( γ x )). Since the factor2 cosh( hL ( γ x )) appears in the expression, it must be corrected by the hyperbolictangent.Finally, the expression that involves S (3) x S (3) y has two terms, corresponding towhether ( x, 0) and ( y, 0) find themselves in the same cycle or not. In the first case,we get cosh( hL ( γ xy )), but in the second case we get sinh( hL ( γ x )) sinh( hL ( γ y )),which eventually gives the hyperbolic tangents. (cid:3) It is convenient to rewrite the cycle weights somewhat. Using 2 cosh( hL ( γ )) =e hL ( γ ) (1 + e − hL ( γ ) ) and (cid:80) γ ∈C ( ω ) L ( γ ) = β |V| , the relevant probability measurefor the cycle representation can be written P cyclesΛ ,β,h ( dω ) = Z ferroΛ (2 β, h ) − e − β |E| + βh |V| d ρ E ,β ( dω ) (cid:89) γ ∈C ( ω ) (cid:0) − hL ( γ ) (cid:1) . (3.11) EISENBERG MODELS AND THEIR PROBABILISTIC REPRESENTATIONS 189 This form makes it easier to see the effect of the external field h ≥ 0. Noticethat the product over cycles simplifies to 2 |C ( ω ) | when the external field strengthvanishes (i.e. h = 0). Then, in terms of the cycle model, the expectation of thespin operators and correlations are given by (cid:104) S (3) x (cid:105) Λ , β,h = E cyclesΛ ,β,h (cid:0) tanh( hL ( γ x )) (cid:1) (3.12) and (cid:104) S (3) x S (3) y (cid:105) Λ , β,h = P cyclesΛ ,β,h ( γ x = γ y )+ E cyclesΛ ,β,h (cid:2) γ x (cid:54) = γ y tanh( hL ( γ x )) tanh( hL ( γ y )) (cid:3) . (3.13) In the case h = 0, we see that (cid:104) S (3) x (cid:105) Λ , β, = 0, as already noted from the spin flipsymmetry, and (cid:104) S (3) x S (3) y (cid:105) Λ , β, = P cyclesΛ ,β,h ( γ x = γ y ) . (3.14) That is, the spin-spin correlation of two sites x and y is proportional to the proba-bility that the sites lie in the same cycle. The antiferromagnetic model only differs from the ferromagnetic model by a sign,but this leads to deep changes. As the transposition operator now carries a negativesign in the Hamiltonian, one possibility would be to turn the measure correspondingto (3.11) into a signed measure, with an extra factor ( − k where k = k ( ω ) isthe number of transpositions. That would mean descending from the heights ofprobability theory to... well, to measure theory. This fate can fortunately beavoided thanks to the observations of Aizenman and Nachtergaele [ ].Their representation is restricted to bipartite graphs. A graph is bipartite ifthe set of vertices V can be partitioned into two sets V A and V B such that edgesonly connect the A set to the B set: { x, y } ∈ E = ⇒ ( x, y ) ∈ V A × V B or ( x, y ) ∈ V B × V A . (3.15) This class contains many relevant cases, such as finite boxes in Z d ; periodic bound-ary conditions are allowed provided the side lengths are even. In the following, weuse the notation ( − x = (cid:40) x ∈ V A , − x ∈ V B . (3.16) Instead of the transposition operator, we consider the projection operator P (0) xy onto the singlet state described in Lemma 2.1. Its action on the basis is P (0) xy | a, a (cid:105) = 0 , P (0) xy | a, − a (cid:105) = | a, − a (cid:105) − | − a, a (cid:105) , (3.17) for all a = ± . Further, it follows from Lemma 2.1 that (cid:126)S x · (cid:126)S y = Id { x,y } − P (0) xy . (3.18) Recall that L ( ω ) is the set of loops of ω . Let γ x denote the loop that contains ( x, t = 0 plane. Also,it is not the lengths of the loops which are important but their winding number w ( γ ). 90 C. GOLDSCHMIDT, D. UELTSCHI, AND P. WINDRIDGE Theorem 3.4 (Aizenman-Nachtergaele’s representation of the antiferromagnet) . Assume that Λ is a bipartite graph. The partition function, the average magnetiza-tion and the two-point correlation function have the following expressions. Z antiΛ (2 β, h ) = e − β |E| (cid:90) d ρ E ,β ( ω ) (cid:89) γ ∈L ( ω ) (cid:0) βhw ( γ )) (cid:1) , Tr S (3) x e − βH antiΛ ,h = ( − x e − β |E| (cid:90) d ρ E ,β ( ω ) tanh( βhw ( γ x )) × (cid:89) γ ∈L ( ω ) (cid:0) βhw ( γ )) (cid:1) , Tr S (3) x S (3) y e − βH antiΛ ,h = ( − x ( − y e − β |E| (cid:90) d ρ E ,β ( ω ) × (cid:89) γ ∈L ( ω ) (cid:0) βhw ( γ )) (cid:1) × (cid:40) if γ x = γ y , tanh( βhw ( γ x )) tanh( βhw ( γ y )) if γ x (cid:54) = γ y . When h = 0, we get the simpler factor 2 |L ( ω ) | . 12 12 1212 12 12 1212 12 1212 12 12 12 Figure 3. Each loop is characterized by a given spin, but thevalues alternate according to whether the site belongs to the A orB sublattice. Proof. As before, we expand the partition function using Corollary 3.2 andwe get Z antiΛ (2 β, h ) = e − β |E| Tr e β (2 hM Λ + (cid:80) e P (0) e ) = e − β |E| (cid:90) d ρ E ,β ( ω ) (cid:88) s ( V ) (cid:104) s ( V ) | e t hM Λ P (0) e . . . P (0) e k e β − t k ) hM Λ | s ( V ) (cid:105) , (3.19) where ( e , t ) , . . . , ( e k , t k ) are the times and the edges of ω . Notice thate tM Λ | s ( V ) (cid:105) = e t (cid:104) s ( V ) | M Λ | s ( V ) (cid:105) | s ( V ) (cid:105) . (3.20) In Dirac’s notation, the resolution of the identity is Id V = (cid:88) s ( V ) ∈{− , } V | s ( V ) (cid:105)(cid:104) s ( V ) | . (3.21) EISENBERG MODELS AND THEIR PROBABILISTIC REPRESENTATIONS 191 We insert it on the right of each operator P (0) e and we obtain Z antiΛ (2 β, h ) = e − β |E| (cid:90) d ρ E ,β ( ω ) (cid:88) s ( V )1 ,...,s ( V ) k e t h (cid:104) s ( V )1 | M Λ | s ( V )1 (cid:105) (cid:104) s ( V )1 | P (0) e | s ( V )2 (cid:105)× e t − t ) h (cid:104) s ( V )2 | M Λ | s ( V )2 (cid:105) (cid:104) s ( V )2 | P (0) e | s ( V )3 (cid:105) . . . (cid:104) s ( V ) k | P (0) e k | s ( V )1 (cid:105) e β − t k ) h (cid:104) s ( V )1 | M Λ | s ( V )1 (cid:105) . (3.22) Let us now observe that this long expression can be conveniently expressed in thelanguage of loops. We can interpret ω and s ( V )1 , . . . , s ( V ) k as a spin configuration s in V × [0 , β ] per . It is constant in time except possibly at ( e i , t i ). By (3.17), theproduct (cid:104) s ( V )1 | P (0) e | s ( V )2 (cid:105) . . . (cid:104) s ( V ) k | P (0) e k | s ( V )1 (cid:105) differs from 0 iff the value of ( − x s ( x, t ) is constant on each loop (see illustrationin Figure 3). In this case, its value is ± 1, as each bridge contributes +1 if the spinsare constant, and − a and b , the factor is( − a − b = e i πa e − i πb . (3.23) Looking at the loop γ with spin a , there is a factor e i πa for each jump A → B (ofthe form (cid:112)(cid:113) ) and a factor e − i πa for each jump B → A (of the form (cid:120)(cid:121) ). Since thereis an identical number of both types of jumps, these factors precisely cancel.The producte t h (cid:104) s ( V )1 | M Λ | s ( V )1 (cid:105) e t − t ) h (cid:104) s ( V )2 | M Λ | s ( V )2 (cid:105) . . . e β − t k ) h (cid:104) s ( V )1 | M Λ | s ( V )1 (cid:105) also factorizes according to loops. The contribution of a loop γ with spin a ise hL A ( γ ) a − hL B ( γ ) a , where L A , L B are the vertical lengths of γ on the A and Bsublattices. We have L A ( γ ) − L B ( γ ) = βw ( γ ) . (3.24) The contribution is therefore e βhw ( γ ) a . Summing over a = ± , we get the hyper-bolic cosine of the expression for the partition function of Theorem 3.4.The expression that involves S (3) x is similar; the only difference is that theloop that contains ( x, 0) contributes ( − x sinh( βhw ( γ )) instead of 2 cosh( βhw ( γ )),hence the hyperbolic tangent. Finally, the expression that involves S (3) x S (3) y issimilar but we need to treat separately the cases where ( x, 0) and ( y, 0) belong ordo not belong to the same loop. (cid:3) 4. Thermodynamic limit and phase transitions Phase transitions are cooperative phenomena where a small change of the exter-nal parameters results in drastic alterations in the properties of the system. Therewas some confusion in the early days of statistical mechanics as to whether the for-malism contained the possibility of describing phase transitions, as all finite volumequantities are smooth. It was eventually realized that the proper formalism involvesa thermodynamic limit where the system size tends to infinity, in such a way thatthe local behavior remains largely unaffected. The proofs of the existence of ther-modynamic limits were fundamental contributions to the mathematical theory ofphase transitions, and they were pioneered by Fisher and Ruelle in the 1960’s; see[ ] for more references. 92 C. GOLDSCHMIDT, D. UELTSCHI, AND P. WINDRIDGE We show that the free energy converges in the thermodynamic limit alonga sequence of boxes in Z d of increasing size (Section 4.1). We discuss variouscharacterizations of ferromagnetic phase transitions in Section 4.2, and magneticlong-range order in Section 4.3. In Section 4.4 we consider the relations betweenthe magnetization in the quantum models and the lengths of the cycles and loops. Despite our professed intention to treat arbi-trary graphs, we now restrict ourselves to a very specific case, namely that of asequence of cubes in Z d whose side lengths tends to infinity. Since F Λ ( β, h ) scaleslike the volume of the system, we define the mean free energy f Λ to be f Λ ( β, h ) = 1 |V| F Λ ( β, h ) . (4.1) We consider the sequence of graphs Λ n = ( V n , E n ) where V n = { , . . . , n } d and E n is the set of nearest-neighbors, i.e., { x, y } ∈ E n iff (cid:107) x − y (cid:107) = 1. Theorem 4.1 (Thermodynamic limit of the free energy) . The sequence of functions ( f Λ n ( β, h )) n ≥ converges pointwise to a function f ( β, h ) , uniformly on compact sets. r nm Figure 4. The large box of size n is decomposed in k d boxes ofsize m ; there are no more than drn d − remaining sites in the darkerarea. Proof. We consider the ferromagnetic model, but the modifications for theantiferromagnetic model are straightforward. We use a subadditive argument. No-tice that the inequality Tr e A + B ≥ Tr e A holds for all self-adjoint operators A, B with B ≥ 0. (This follows e.g. from the minimax principle, or from Klein’s in-equality.) We rewrite the Hamiltonian so as to have only positive definite terms.Namely, let h x,y = (cid:126)S x · (cid:126)S y + Id . (4.2) Then Z Λ ( β, h ) = e − β |E| Tr exp (cid:16) β (cid:88) { x,y }∈E h x,y + βh (cid:88) x ∈V S (3) x (cid:17) . (4.3) Let m, n, k, r be integers such that n = km + r and 0 ≤ r < m . The box V n isthe disjoint union of k d boxes of size m , and of some remaining sites (fewer than drn d − ); see Figure 4 for an illustration. We get an inequality for the partition EISENBERG MODELS AND THEIR PROBABILISTIC REPRESENTATIONS 193 function in Λ n by dismissing all h x,y where { x, y } are not inside a single box of size m . The boxes V m become independent, and Z Λ n ( β, h ) ≥ e − β |E n | (cid:104) Tr H ( V m ) exp (cid:16) β (cid:88) { x,y }∈E m h x,y + βh (cid:88) x ∈V m S (3) x (cid:17)(cid:105) k d = [ Z Λ m ( β, h )] k d e − β |E n | e k d β |E m | . (4.4) We have neglected the contribution of e βhS (3) x for x outside the boxes V m , which ispossible because their traces are greater than 1. It is not hard to check that |E n | ≤ k d |E m | + k d dm d − + d rn d − . (4.5) We then obtain a subbaditive relation for the free energy, up to error terms thatwill soon disappear: f Λ n ( β, h ) ≤ ( km ) d n d f Λ m ( β, h ) + 3 k d dm d − n d + 3 d r n . (4.6) Then, since kmn → n → ∞ ,lim sup n →∞ f Λ n ( β, h ) ≤ f Λ m ( β, h ) + 3 d m . (4.7) Taking the lim inf over m in the right side, we see that it is larger or equal to thelim sup, and so the limit necessarily exists.Uniform convergence on compact intervals follows from Lemma 2.4 (which im-plies that ( f Λ n ) is equicontinuous) and the Arzel`a-Ascoli theorem (see e.g. Theorem4.4 in Folland [ ]). (cid:3) Corollary 4.2 (Thermodynamic limit with periodic boundary conditions) . Let (Λ per n ) be the sequence of cubes in Z d of size n with periodic boundary conditionsand nearest-neighbor edges. Then ( f Λ per n ( β, h )) n ≥ converges pointwise to the samefunction f ( β, h ) as in Theorem 4.1, uniformly on compact sets. This follows from | f Λ per n ( β, h ) − f Λ n ( β, h ) | ≤ d n , which is not too hard to prove,and Theorem 4.1. In statistical physics, an order pa-rameter is a quantity which allows one to identify a phase, typically because itvanishes in all phases except one. The relevant order parameter here is the mag-netization, which is expected to be zero at high temperatures and positive at lowtemperatures. There are actually three natural definitions for the magnetization;we show below that the first two are equivalent, and that the last one is smallerthan the first two. • Thermodynamic magnetization . This is equal to (the negative of) theright-derivative of f ( β, h ) with respect to h . We are looking for a jumpin the derivative, which is referred to as a first-order phase transition. m ∗ th ( β ) = − lim h → f ( β, h ) − f ( β, h . (4.8) (The limit exists because f is concave.) 94 C. GOLDSCHMIDT, D. UELTSCHI, AND P. WINDRIDGE • Residual magnetization . Imagine placing the ferromagnet in an exter-nal magnetic field, so that it becomes magnetized. Now gradually turnoff the external field. Does the system still display global magnetization?Mathematically, the relevant order parameter is m ∗ res ( β ) = lim h → lim inf n →∞ n d (cid:104) M Λ n (cid:105) Λ n ,β,h . (4.9) (We see below that the lim inf can be replaced by the lim sup withoutaffecting m ∗ res . The limit over h exists because (cid:104) M Λ n (cid:105) is the derivative ofa concave function, and it is therefore monotone.) • Spontaneous magnetization at h = 0. Since (cid:104) M Λ (cid:105) = 0 (because of thespin flip symmetry), we rather consider m ∗ sp ( β ) = lim inf n →∞ n d (cid:104)| M Λ n |(cid:105) Λ n ,β, . (4.10) Here, | M Λ | denotes the absolute value of the matrix M Λ .A handier quantity, however, is the expectation of M , which can be expressed interms of the two-point correlation function, see below. It is equivalent to m ∗ sp inthe sense that both are zero or both differ from zero: Lemma 4.3. (cid:10) | M Λ ||V| (cid:11) ,β, ≤ (cid:10)(cid:0) M Λ |V| (cid:1) (cid:11) Λ ,β, ≤ (cid:10) | M Λ ||V| (cid:11) Λ ,β, . Proof. For the first inequality, use | M Λ | = | M Λ | Id and then the Cauchy-Schwarz inequality (2.24). For the second inequality, observe that | M Λ | ≤ |V| Id implies that M ≤ |V|| M Λ | , and use the fact that the Gibbs state is a positivelinear functional. (cid:3) f h Figure 5. Qualitative graphs of the free energy f ( β, h ) as a func-tion of h , for β large (top) and β small (bottom). Proposition 4.4. The three order parameters above are related as follows: m ∗ th ( β ) = m ∗ res ( β ) ≥ m ∗ sp ( β ) . Proof of m ∗ th = m ∗ res . We prove that whenever f n is a sequence of differen-tiable concave functions that converge pointwise to the (necessarily concave) func-tion f , then D + f (0) = lim h → lim sup n →∞ f (cid:48) n ( h ) = lim h → lim inf n →∞ f (cid:48) n ( h ) . (4.11) EISENBERG MODELS AND THEIR PROBABILISTIC REPRESENTATIONS 195 Up to the signs, the left side is equal to m ∗ th and the right side to m ∗ res and weobtain the identity in Proposition 4.4. The proof of (4.11) follows from the generalproperties lim sup i (cid:0) inf j a ij (cid:1) ≤ inf j (cid:0) lim sup i a ij (cid:1) , lim inf i (cid:0) sup j a ij (cid:1) ≥ sup j (cid:0) lim inf i a ij (cid:1) , (4.12) and from the following expressions for left- and right-derivatives of concave func-tions: D − f ( h ) = inf s> f ( h ) − f ( h − s ) s , D + f ( h ) = sup s> f ( h + s ) − f ( h ) s . (4.13) With these observations, the proof is straightforward. For h > D + f (0) ≥ D − f ( h ) = inf s> lim sup n →∞ f n ( h ) − f n ( h − s ) s ≥ lim sup n →∞ f (cid:48) n ( h ) ≥ lim inf n →∞ f (cid:48) n ( h ) ≥ sup s> lim inf n →∞ f n ( h + s ) − f n ( h ) s = D + f ( h ) . (4.14) Since right-derivatives of concave functions are right-continuous, the last term con-verges to D + f (0) as h → (cid:3) Proof of m ∗ res ≥ m ∗ sp . Let h > 0, and let { ϕ j } be an orthonormal set ofeigenvectors of H Λ n , and M Λ n with eigenvalues e j and m j , respectively. Becauseof the spin flip symmetry, we have (cid:104) M Λ n (cid:105) Λ n ,β,h = (cid:80) j : m j > m j e − βe j (cid:0) e βhm j − e − βhm j (cid:1)(cid:80) j : m j > e − βe j (cid:0) e βhm j + e − βhm j (cid:1) + (cid:80) j : m j =0 e − βe j ≥ (cid:80) j : m j > m j e − βe j + βhm j (cid:0) − e − βhm j (cid:1) (cid:80) j : m j > e − βe j + βhm j + (cid:80) j : m j =0 e − βe j . (4.15) After division by n d , we only need to consider those j with m j ∼ n d , in which casee − βhm j ≈ 0. We can therefore replace the parenthesis by 1 in the limit n → ∞ .On the other hand, consider the function G n ( h ) = β log Tr e − βH Λ n, + βh | M Λ n | . Onecan check that it is convex in h , see (2.22), so G (cid:48) n ( h ) ≥ G (cid:48) n (0). Its derivative canbe expanded as above, so that G (cid:48) n ( h ) = (cid:80) j | m j | e − βe j + βh | m j | (cid:80) j e − βe j + βh | m j | . (4.16) This is equal to twice the second line of (4.15) (without the parenthesis). Then m ∗ res ( β ) ≥ lim h → lim inf n →∞ n d G (cid:48) n ( h ) ≥ lim inf n →∞ n d G (cid:48) n (0) = m ∗ sp ( β ) . (4.17) (cid:3) While ferromagnets favor align-ment of the spins, antiferromagnets favor staggered phases, where spins are alignedon one sublattice and aligned in the opposite direction on the other sublattice. Theexternal magnetic field does not play much of a rˆole. One could mirror the ferro-magnetic situation by introducing a non-physical staggered magnetic field of thekind h (cid:80) x ∈V ( − x S (3) x , which would lead to counterparts of the order parameters 96 C. GOLDSCHMIDT, D. UELTSCHI, AND P. WINDRIDGE m ∗ th and m ∗ res . We content ourselves with turning off the external magnetic field,i.e. setting h = 0, and with looking at magnetic long-range order. For x, y ∈ V , weintroduce the correlation function σ Λ ,β ( x, y ) = ( − x ( − y (cid:104) S (3) x S (3) y (cid:105) Λ ,β, . (4.18) One question is whether σ ( β ) = lim inf n →∞ |V n | (cid:88) x,y ∈V n σ Λ ,β ( x, y ) (4.19) differs from 0. A related question is whether the correlation function does not decayto 0 as the distance between x and y tends to infinity. One says that the systemexhibits long-range order if this happens.In Z d and for β large enough, it is well-known that there is no long-range orderand that the correlation function decays exponentially in (cid:107) x − y (cid:107) . Long-range orderis expected in dimension d ≥ d = 1 , 2. This is discussed in more detailin Section 5. In this section, weclarify the relations between the order parameters of the quantum systems and thenature of cycles and loops. This yields probabilistic interpretations for the quantumresults. We introduce two quantities, which apply simultaneously to cycles andloops. Recall that γ x denotes either the cycle that contains ( x, 0) in the cyclemodel, or the loop that contains ( x, 0) in the loop model. We write P and E for P cycles and E loop when equations hold in both cases. • The fraction of vertices in infinite cycles/loops: η ∞ ( β, h ) = lim K →∞ lim inf n →∞ n d E Λ n ,β,h (cid:0) { x ∈ V n : L ( γ x ) > K } (cid:1) . (4.20) • The fraction of vertices in macroscopic cycles/loops: η macro ( β, h ) = lim ε → lim inf n →∞ n d E Λ n ,β,h (cid:0) { x ∈ V n : L ( γ x ) > εn d } (cid:1) . (4.21) It is clear that η ∞ ( β, h ) ≥ η macro ( β, h ). These two quantities relate to magnetiza-tion and long-range order as follows. The first two statements deal with cycles andthe third statement deals with loops. Proposition 4.5. (a) m ∗ res (2 β ) ≥ lim h → η ∞ ( β, h ) . (b) m ∗ sp (2 β ) > ⇐⇒ η macro ( β, > . (c) σ (2 β ) > ⇐⇒ η macro ( β, > . Proof. Let m (2 β, h ) = lim inf n →∞ (cid:104) S (3)0 (cid:105) Λ n , β,h . (4.22) We use tanh x ≥ tanh K · x>K , which holds for any K , and Theorem 3.3, so as toget m (2 β, h ) ≥ tanh( hK ) lim inf n →∞ P cyclesΛ n ,β,h ( L ( γ ) > K ) . (4.23) Taking K → ∞ , we get m (2 β, h ) ≥ η ∞ ( β, h ). We now take h → 0+ to obtain (a).For (b), we observe that, since the vertices of Λ n are exchangeable,1 n d (cid:104) M n (cid:105) Λ n , β, = 12 β E cyclesΛ n ,β, (cid:16) L ( γ ) n d (cid:17) . (4.24) EISENBERG MODELS AND THEIR PROBABILISTIC REPRESENTATIONS 197 It follows from Lemma 4.3 that m ∗ sp (2 β ) > ⇐⇒ lim inf n →∞ E cyclesΛ n ,β, (cid:16) L ( γ ) n d (cid:17) > . (4.25) On the other hand, we have η macro ( β, 0) = lim ε → lim inf n →∞ P cyclesΛ n ,β, (cid:16) L ( γ ) n d > ε (cid:17) . (4.26) The result is then clear.The claim (c) is identical to (b), with loops instead of cycles. (cid:3) It should be possible to extend Proposition 4.5 (a) so that m ∗ res ( β ) > ⇔ η ∞ ( β, > 0. This suggests that m ∗ th and m ∗ res are related to the existence ofinfinite cycles, while m ∗ sp is related to the occurrence of macroscopic cycles. Thequestion is then whether there exists a phase in which a positive fraction of verticesbelongs to mesoscopic cycles or loops. Such a phase could have something to do withthe Berezinski˘ı-Kosterlitz-Thouless transition [ 7, 36 ], which has been rigorouslyestablished in the classical XY model [ ]. It is not expected in the Heisenbergmodel, though. The Mermin-Wagner theorem (Section 5.1) rules out any kind ofinfinite cycles or loops in one and two dimensions. 5. Rigorous results for the quantum models Quantum lattice systems have seen a considerable amount of study in the pastdecades, and the effort is not abating. Physicists are interested in properties of theground state (i.e., the eigenvector of the Hamiltonian with lowest eigenvalue), indynamical behavior, and in the existence and nature of phase transitions. Out ofmany results, we only discuss two in this section, which have been chosen becauseof their direct relevance to the understanding of the cycle and loop models: theMermin-Wagner theorem concerning the absence of spontaneous magnetization inone and two dimensions, and the theorem of Dyson, Lieb, and Simon concerningthe existence of long-range order in the antiferromagnetic model. This fundamental result of condensed mat-ter physics states that a continuous symmetry cannot be broken in one and twodimensions [ ]. In particular, there is no spontaneous magnetization or long-rangeorder in Heisenberg models. Theorem 5.1. Let (Λ per n ) n ≥ be the sequence of cubic boxes in Z d with periodicboundary conditions. For d = 1 or 2, and for any β ∈ [0 , ∞ ) , m ∗ res ( β ) = 0 . By Proposition 4.4, all three ferromagnetic order parameters are zero, and thereare no infinite cycles by Proposition 4.5 in the cycle model that corresponds to theHeisenberg ferromagnet. The theorem can also be stated for the staggered magneticfield discussed in Section 4.3. One could establish antiferromagnetic counterpartsto Lemma 4.3 and Proposition 4.4, and therefore prove that η ∞ ( β ) is also zero inthe loop model that corresponds to the Heisenberg antiferromagnet.An open question is whether the theorem can be extended to more generalmeasures of the form ϑ |C ( ω ) | d ρ E ,β ( ω ) and ϑ |L ( ω ) | d ρ E ,β ( ω ) 98 C. GOLDSCHMIDT, D. UELTSCHI, AND P. WINDRIDGE (up to normalization), for values of ϑ other than ϑ = 2. The case 3 |L ( ω ) | can actuallybe viewed as the representation of a model with Hamiltonian − (cid:80) { x,y }∈E ( (cid:126)S x · (cid:126)S y ) (see [ ]) and the Mermin-Wagner theorem certainly holds in that case.The theorem may not apply when ϑ is too large, and the system is in a phasewith many loops, similar to the one studied in [ ].We present the standard proof [ ] that is based on Bogolubov’s inequality. Proposition 5.2 (Bogolubov’s inequality) . Let β > and A, B, H be operators ona finite-dimensional Hilbert space, with H self-adjoint. Then (cid:12)(cid:12) Tr [ A, B ] e − βH (cid:12)(cid:12) ≤ β Tr ( AA ∗ + A ∗ A ) e − βH Tr (cid:2) [ B, H ] , B ∗ (cid:3) e − βH . Proof. We only sketch the proof; see [ ] for more details. Let { ϕ i } be anorthonormal set of eigenvectors of H and { e i } the corresponding eigenvalues. Weintroduce the following inner product:( A, C ) = (cid:88) i,j : e i (cid:54) = e j (cid:104) ϕ i , A ∗ ϕ j (cid:105)(cid:104) ϕ j , Cϕ i (cid:105) e − βe j − e − βe i e i − e j . (5.1) One can check that ( A, A ) ≤ β Tr ( AA ∗ + A ∗ A ) e − βH . (5.2) We choose C = [ B ∗ , H ], and we check thatTr [ A, B ] e − βH = ( A, C ) (5.3) and Tr (cid:2) [ B, H ] , B ∗ (cid:3) e − βH = ( C, C ) . (5.4) Inserting (5.3) and (5.4) in the Cauchy-Schwarz inequality of the inner product(5.1), and using (5.2), we get Bogolubov’s inequality. (cid:3) Proof of Theorem 5.1. Let m n ( β, h ) = n − d (cid:104) M Λ n (cid:105) Λ n ,β,h . Let S ( ± ) x = √ ( S (1) x ± i S (2) x ) . (5.5) One easily checks that [ S (+) x , S ( − ) y ] = S (3) x δ x,y . (5.6) It is convenient to label the sites of Λ per n as follows V n = { x ∈ Z d : − n < x i ≤ n , i = 1 , . . . , d } . (5.7) E n is again the set of nearest-neighbors in V n with periodic boundary conditions.For k ∈ πn V n , we introduce S ( · ) ( k ) = 1 n d/ (cid:88) x ∈V n e − i kx S ( · ) x , (5.8) where kx denotes the inner product in R d . Then, using (5.6), (cid:104) [ S (+) ( k ) , S ( − ) ( − k )] (cid:105) Λ n ,β,h = 1 n d (cid:88) x,y ∈V n e − i kx e i ky (cid:104) [ S (+) x , S ( − ) y ] (cid:105) Λ n ,β,h = m n ( β, h ) . (5.9) EISENBERG MODELS AND THEIR PROBABILISTIC REPRESENTATIONS 199 This will be the left side of Bogolubov’s inequality. For the right side, tedious butstraightforward calculations (expansions, commutation relations) give (cid:10)(cid:2) [ S (+) ( k ) , H Λ n ] , S ( − ) ( − k ) (cid:3)(cid:11) Λ n ,β,h = 2 n d (cid:88) x,y : { x,y }∈E n (1 − e i k ( x − y ) ) (cid:10) S ( − ) x S (+) y + S (3) x S (3) y (cid:11) Λ n ,β,h + hm n ( β, h ) . (5.10) Despite appearances, this expression is real and positive for any k , as can be seenfrom (5.4). We get an upper bound by adding the same quantity, but with − k .This yields4 n d (cid:88) x,y : { x,y }∈E n (1 − cos k ( x − y )) (cid:10) S ( − ) x S (+) y + S (3) x S (3) y (cid:11) Λ n ,β,h + 2 hm n ( β, h ) . From Lemma 2.1, we have (cid:12)(cid:12)(cid:10) S ( − ) x S (+) y + S (3) x S (3) y (cid:11) Λ n ,β,h (cid:12)(cid:12) = (cid:12)(cid:12) (cid:104) (cid:126)S x · (cid:126)S y (cid:105) Λ n ,β,h (cid:12)(cid:12) ≤ . (5.11) Let us now introduce the “dispersion relation” of the lattice: ε ( k ) = d (cid:88) i =1 (1 − cos k i ) . (5.12) Inserting all of this into Bogolubov’s inequality, we get m n ( β, h ) ε ( k ) + 2 | hm n ( β, h ) | ≤ β (cid:10) S (+) ( k ) S ( − ) ( − k ) + S ( − ) ( − k ) S (+) ( k ) (cid:11) Λ n ,β,h . (5.13) Summing over all k ∈ πn V n , and using (cid:80) k e − i k ( x − y ) = δ x,y , we have (cid:88) k (cid:10) S (+) ( k ) S ( − ) ( − k ) + S ( − ) ( − k ) S (+) ( k ) (cid:11) Λ n ,β,h = (cid:88) x ∈V n (cid:10) S (+) x S ( − ) x + S ( − ) x S (+) x (cid:11) Λ n ,β,h = n d . (5.14) Then m n ( β, h ) n d (cid:88) k ∈ πn V n ε ( k ) + 2 | hm n ( β, h ) | ≤ β. (5.15) As n → ∞ , we get a Riemann integral, m ( β, h ) π ) d (cid:90) [ − π,π ] d d k ε ( k ) + 2 | hm n ( β, h ) | ≤ β. (5.16) Since ε ( k ) ≈ k around k = 0, the integral diverges when h → 0, and so m ( β, h )must go to 0. (cid:3) Notice that the integral remains finite for d ≥ 3; the argument only applies to d = 1 , 00 C. GOLDSCHMIDT, D. UELTSCHI, AND P. WINDRIDGE Fol-lowing the proof of Fr¨ohlich, Simon and Spencer of a phase transition in the classicalHeisenberg model [ ], Dyson, Lieb and Simon proved the existence of long-rangeorder in several quantum lattice models, including the antiferromagnetic quantumHeisenberg model in dimensions d ≥ ]. Further observations of Neves andPerez [ ], and of Kennedy, Lieb and Shastry [ ], imply that long-range order ispresent for all d ≥ These articles use the “reflection positivity” method, whichwas systematized and extended in [ 22, 23 ]. We recommend the Prague notes ofT´oth [ ] and Biskup [ ] for excellent introductions to the topic. See also thenotes of Nachtergaele [ ].Recall the definition of σ in Eq. (4.19). Theorem 5.3 (Dyson-Lieb-Simon) . Let (Λ per n ) be the sequence of cubic boxes in Z d , d ≥ , with even side lengths and periodic boundary conditions. There exists β < ∞ such that, for all β > β , the Heisenberg antiferromagnet has long-rangeorder, σ ( β ) > . Clearly, this theorem has remarkable consequences for the loop model withweights 2 |L ( ω ) | . Indeed, there are macroscopic loops, η macro ( β, > 0, providedthat β is large enough.Despite many efforts and false hopes, there is no corresponding result for theHeisenberg ferromagnet, and hence for the cycle model.The proof of Theorem 5.3 for d ≥ ] (see also [ 22, 49 ]for useful clarifications). In the remainder of this section we explain how to usethe observations of [ ] and [ ] in order to extend the result to dimensions d = 3and d = 4. As these articles deal with ground state properties rather than positivetemperatures, some modifications are needed. We warn the readers that this partof the notes is not really self-contained.Recall the definitions of the operators S ( · ) ( k ) in Eq. (5.8). We need theDuhamel two-point function, which is reminiscent of the Duhamel formula of Propo-sition 3.1.( S ( j ) ( k ) , S ( j ) ( − k )) Λ ,β, = 1 Z Λ ( β, (cid:90) Tr e − sβH Λ , S ( j ) ( k ) e − (1 − s ) βH Λ , S ( j ) ( − k )d s. (5.17) Recall also the definition of ε ( k ) in (5.12), and let (cid:126)π = ( π, . . . , π ) ∈ R d . We have ε ( k − (cid:126)π ) = d (cid:88) i =1 (1 + cos k i ) . (5.18) Let e n ( β ) denote the negative of the mean energy per site, i.e., e n ( β ) = − (cid:68) H Λ , n d (cid:69) Λ ,β, . (5.19) One can show that e n ( β ) is nonnegative, increasing with respect to β , and that itconverges pointwise as n → ∞ .The main result of reflection positivity is the following “Gaussian domination”. Proposition 5.4. If k ∈ πn V n and k (cid:54) = (cid:126)π , we have We are indebted to the anonymous referee for pointing this out and for clarifying this to us.The following explanation is essentially taken from the referee’s report. EISENBERG MODELS AND THEIR PROBABILISTIC REPRESENTATIONS 201 (a) ( S ( j ) ( k ) , S ( j ) ( − k )) Λ n ,β, ≤ ε ( k − (cid:126)π ) , (b) (cid:104) S ( j ) ( k ) S ( j ) ( − k ) (cid:105) Λ n ,β, ≤ (cid:16) e n ( β )6 d (cid:17) / (cid:16) ε ( k ) ε ( k − (cid:126)π ) (cid:17) / + 32 βε ( k − (cid:126)π ) . Sketch proof. The claim (a) can be found in [ ], Theorem 6.1. The claim(b) follows from Eqs (3), (5), and (6) of [ ], and from the relation (cid:88) j =1 (cid:10)(cid:2) S ( j ) ( k ) , [ H Λ , , S ( j ) ( − k )] (cid:3)(cid:11) Λ ,β, = d ε ( k ) e n ( β ) . (5.20) This is Eq. (55) in [ ]. (cid:3) Next, let σ n ( β ) = 1 n d (cid:88) x,y ∈V n ( − x ( − y (cid:104) S (3) x S (3) y (cid:105) Λ ,β, . (5.21) Then σ ( β ) = lim inf n σ n ( β ), and the goal is to show that it differs from zero. For k = (cid:126)π , we have (cid:104) S (3) ( (cid:126)π ) S (3) ( − (cid:126)π ) (cid:105) Λ ,β, = n d σ n ( β ) . (5.22) Kennedy, Lieb and Shastry [ ] have proposed the following sum rule, which im-proves on the original one used in [ 24, 17 ]:1 n d (cid:88) k ∈ πn V n (cid:104) S (3) ( k ) S (3) ( − k ) (cid:105) Λ ,β, cos k i = (cid:104) S (3)0 S (3) e i (cid:105) Λ ,β, , (5.23) where e i denotes the neighbor of the origin in the i th direction. Because of thesymmetries of V n (translations and lattice rotations), we have (cid:104) S (3)0 S (3) e i (cid:105) Λ ,β, = − e n ( β )3 d . (5.24) The sum rule can be rewritten as e n ( β )3 d = σ n ( β ) + 1 dn d (cid:88) k ∈ πn V n k (cid:54) = (cid:126)π (cid:104) S (3) ( k ) S (3) ( − k ) (cid:105) Λ ,β, (cid:16) − d (cid:88) i =1 cos k i (cid:17) . (5.25) By Proposition 5.4 (b), we have e n ( β )3 d ≤ σ n ( β ) + e n ( β ) / (6 d ) / d n d (cid:88) k ∈ πn V n k (cid:54) = (cid:126)π (cid:16) ε ( k ) ε ( k − (cid:126)π ) (cid:17) / (cid:16) − d (cid:88) i =1 cos k i (cid:17) + + 32 dβ n d (cid:88) k ∈ πn V n k (cid:54) = (cid:126)π ε ( k − (cid:126)π ) (cid:16) − d (cid:88) i =1 cos k i (cid:17) + . (5.26) As n → ∞ , with e ( β ) = lim n e n ( β ), we get e ( β )3 d ≤ σ ( β )+ e ( β ) / (6 d ) / d I ( d )+ 32 dβ π ) d (cid:90) [ − π,π ] d ε ( k − (cid:126)π ) (cid:16) − d (cid:88) i =1 cos k i (cid:17) + d k, (5.27) 02 C. GOLDSCHMIDT, D. UELTSCHI, AND P. WINDRIDGE where I ( d ) = 1(2 π ) d (cid:90) [ − π,π ] d (cid:16) ε ( k ) ε ( k − (cid:126)π ) (cid:17) / (cid:16) − d (cid:88) i =1 cos k i (cid:17) + d k. (5.28) The last integral in (5.27) is finite when d ≥ 3, and this term may be made arbi-trarily small by choosing β large enough. It follows that a sufficient condition for σ ( β ) > β , is thatlim β →∞ e ( β ) / d > d ) / d I ( d ) . (5.29) The integral I ( d ) can be calculated numerically: I (3) = 1 . ... and I (4) =1 . ... It is then enough to show that lim β →∞ e ( β ) > . ... in d = 3 andlim β →∞ e ( β ) > . ... in d = 4. The following lemma allows us to conclude thatlong-range order indeed takes place in d = 3 and d = 4. Lemma 5.5. lim β →∞ e ( β ) ≥ d . Proof. The Gibbs variational principle states that F Λ ( β, h ) ≤ Tr ρH Λ ,h − β S Λ ( ρ ) (5.30) for any operator ρ in H ( V ) such that ρ ≥ ρ = 1. Here, S Λ is the Boltzmannentropy, S Λ ( ρ ) = − Tr ρ log ρ. (5.31) See e.g. Proposition IV.2.5 in [ ] (the setting in [ ] involves a normalized trace,hence there are a few discrepancies between our formulæ and those in the book).It is known that the Gibbs state ρ = Z Λ ( β, h ) − e − βH Λ ,h saturates the inequality,and that the entropy satisfies the bounds0 ≤ S Λ ( ρ ) ≤ |V| log 2 . (5.32) It follows that e ( β ) ≥ − f ( β, − log 2 β . (5.33) In order to get a bound for the free energy, we use (5.30) with the N´eel state Ψ N´eel as a trial state, Ψ N´eel = (cid:79) x ∈ Λ n (cid:12)(cid:12) ( − x (cid:11) . (5.34) With ρ the projector onto Ψ N´eel , we have S Λ n ( ρ ) = 0, and F Λ n ( β, ≤ (cid:104) Ψ N´eel , H Λ n , Ψ N´eel (cid:105) = dn d (cid:104) , − | (cid:126)S x · (cid:126)S y | , − (cid:105) . (5.35) The last inner product is in H x ⊗H y . Using (3.8), we find that it is equal to − . (cid:3) These results do not apply to dimension d = 2 because the last integral in(5.27) is divergent. We already know that the magnetization is zero for all finitevalues of β by the Mermin-Wagner theorem. An important question, which remainsopen to this day, is whether long-range order occurs in the ground state of the two-dimensional antiferromagnet. The last integral in (5.27) disappears if the limit β → ∞ is taken before the infinite volume limit, and the question is whether (5.29) EISENBERG MODELS AND THEIR PROBABILISTIC REPRESENTATIONS 203 is true. Since I (2) = 1 . ... one needs lim β →∞ e ( β ) > . ... . But the limit isexpected to be around 0.67 [ ] and so the method does not apply.In contrast to the antiferromagnet, the ground state of the ferromagnet is triv-ial with full magnetization. If β is taken to infinity in the cycle model for a fixedgraph, the spatial structure is lost and the resulting random permutation has Ewensdistribution (that is, it is weighted by 2 |C| ). Almost all vertices belong to macro-scopic cycles and the cycle lengths are distributed according to the Poisson-Dirichletdistribution PD . 6. Rigorous results for cycle and loop models The cycle and loop representations in Theorems 3.3 and 3.4 are interesting intheir own right and can be studied using purely probabilistic techniques. Withoutthe physical motivation, the external magnetic field is less relevant and more ofan annoyance. We prefer to switch it off. The models in this simpler situationare defined below, with the small generalization that the geometric weight on thenumber of cycles or loops is arbitrary. This is analogous to how, for example,one obtains the random cluster or Fortuin-Kasteleyn representation from the Isingmodel. As usual we suppose that Λ = ( V , E ) is afinite undirected graph. Recall that the Poisson edge measure ρ E ,β is obtained byattaching independent Poisson point processes on [0 , β ] to each edge of E .For each realization ω of the Poisson edge process, we define cycles C ( ω ) andloops L ( ω ) as in § ϑ > P cyclesΛ ,β (d ω ) = Z cyclesΛ ( β ) − ϑ |C ( ω ) | ρ E ,β (d ω ) , P loopsΛ ,β (d ω ) = Z loopsΛ ( β ) − ϑ |L ( ω ) | ρ E ,β (d ω ) , (6.1) where Z ··· Λ ( β ) are the appropriate normalizations. As remarked above, ϑ = 2 is thephysically relevant choice in both these measures.The main question deals with the possible occurrence of cycles or loops ofdiverging lengths. Recall the definitions of the fraction of vertices in infinite cy-cles/loops, η ∞ ( β ), and the fraction of vertices in macroscopic cycles/loops, η macro ( β ), which were defined in Section 4.4. (We drop the dependence in h , since h = 0 here.) In the case where the graph is a cubic box in Z d with periodic bound-ary conditions, and ϑ = 2, the Mermin-Wagner theorem rules out infinite cycles inone and two dimensions, and the theorem of Dyson-Lieb-Simon shows that macro-scopic loops are present in d ≥ 3, provided that the parameter β is sufficientlylarge.It is intuitively clear that there cannot be infinite cycles or loops when β issmall. In Section 6.2 we prove this is indeed the case and give an explicit lowerbound on the critical value of β .The model for ϑ = 1 is known as random stirring or the interchange process.The question of the existence of infinite cycles in this setting has been consideredby several authors. Angel considered the model on regular trees, and proved theexistence of infinite cycles (for β lying in an appropriate interval) when the degree ofthe tree is larger than 5 [ ]. Schramm considered the model on the complete graph 04 C. GOLDSCHMIDT, D. UELTSCHI, AND P. WINDRIDGE and obtained a fairly precise description of the asymptotic cycle length distribution[ ]. We review this important result in Section 6.3. Recently, Alon and Kozmafound a surprising formula for the probability that the permutation is cyclic, usingrepresentation theory [ ]. We consider general graphsΛ = ( V , E ). We let κ denote the maximal degree of the graph, i.e., κ = sup x ∈V |{ y : { x, y } ∈ E}| . Recall that L ( γ x ) denotes the length of the cycle or loop that contains x × { } . Let a be the small parameter a = (cid:40) ϑ − (1 − e − β ) if ϑ ≤ , − e − β if ϑ ≥ . (6.2) in the case of cycles and a = (cid:40) ϑ − (1 − e − β ) if ϑ ≤ , e − β ( e βϑ − 1) if ϑ ≥ . (6.3) in the case of loops. Theorem 6.1. For either the cycle or the loop model, i.e., for either measure in(6.1), we have P Λ ,β ( L ( γ x ) > βk ) ≤ ( a ( κ − − [ aκ (1 − κ ) − κ +1 ] k . for every x ∈ V . Of course, the theorem is useful only if the right-hand side is less than 1, inwhich case large cycles have exponentially small probability. This result is prettyreasonable on the square lattice with ϑ ≤ 1. When ϑ > 1, configurations with manycycles are favored, and the domain should allow for larger β . Our condition doesnot show it. The case ϑ (cid:29) ] with phases ofclosely packed loops. In the case of the complete graph on N vertices and ϑ = 1, themaximal degree is κ = N − β < /N (Erd˝os-R´enyi,[ ]). Using aκ ≤ βN and (1 − κ ) − κ +1 ≤ e , we see that our condition is off by afactor of e .As a consequence of the theorem, we have η ∞ ( β ) = 0 for small enough β . Thisimplies that m ∗ sp ( β ) = σ ( β ) = 0 in the corresponding Heisenberg ferromagnet andantiferromagnet. One could extend the claim so that m ∗ th ( β ) = 0 as well. Proof. Given ω , let G ( ω ) = ( V, E ) denote the subgraph of Λ with edges E = { e i : ( e i , t i ) ∈ ω } , (6.4) and V = ∪ i e i the set of vertices that belong to at least one edge. G ( ω ) can beviewed as the percolation graph of ω , where an edge e is open if at least one bridgeof the form ( e, t ) occurs in ω . Then we denote C x ( ω ) = ( V x , E x ) the connectedcomponent of G ( ω ) that contains x . It is clear that L ( γ x ) ≤ β | V x | for both cyclesand loops. Then, using Markov’s inequality, P Λ ,β ( L ( γ x ) > βk ) ≤ P Λ ,β ( | V x | > k ) ≤ α − k E Λ ,β ( α | V x | ) , (6.5) for any α ≥ G (cid:48) = ( V (cid:48) , E (cid:48) ) of Λ, let φ ( G (cid:48) ) = ϑ −| V (cid:48) | (cid:90) [ G ( ω )= G (cid:48) ] ϑ |C ( ω ) | d ρ E (cid:48) ,β ( ω ) . (6.6) EISENBERG MODELS AND THEIR PROBABILISTIC REPRESENTATIONS 205 By partitioning Ω according to the connected components of G ( ω ), then usingthe fact that ρ E ,β is a product measure over edges and that cycles are containedentirely within connected components, we have E cyclesΛ ,β ( α | V (cid:48) x | ) = (cid:88) C (cid:48) x φ ( C (cid:48) x ) α | V (cid:48) x | (cid:80) G (cid:48) ∩ C (cid:48) x = ∅ φ ( G (cid:48) ) (cid:80) G (cid:48)(cid:48) φ ( G (cid:48)(cid:48) ) (6.7) The first sum is over connected subgraphs C (cid:48) x = ( V (cid:48) x , E (cid:48) x ) of Λ that contain x .The second sum is over subgraphs G (cid:48) = ( V (cid:48) , E (cid:48) ) that are compatible with C (cid:48) x , inthe sense that V (cid:48) ∩ V (cid:48) x = ∅ and V (cid:48) ∪ C (cid:48) x = V . The sum in the denominator is overall subgraphs G (cid:48)(cid:48) = ( V (cid:48)(cid:48) , E (cid:48)(cid:48) ) with V (cid:48)(cid:48) = V .Notice that for any C (cid:48) x , the corresponding compatible graph G (cid:48) = ( V (cid:48) , E (cid:48) ) canbe enlarged to G (cid:48)(cid:48) = ( V , E (cid:48) ) by adding the vertices from V (cid:48) x . The new vertices from V (cid:48) x are all disconnected in G (cid:48)(cid:48) . Thus, if G ( ω ) = G (cid:48)(cid:48) , each vertex in V (cid:48) x necessarilyforms a single cycle of length 1. It follows that φ ( G (cid:48)(cid:48) ) = φ ( G (cid:48) ). Furthermore,different G (cid:48) give rise to different G (cid:48)(cid:48) . So, the ratio in (6.7) is less than 1.Now we claim that φ ( G (cid:48) ) ≤ a | E (cid:48) | (6.8) for any connected G (cid:48) . First consider ϑ ≤ 1. Since G (cid:48) is connected we have | E (cid:48) | ≥| V (cid:48) | − 1. So, ϑ −| V (cid:48) | + |C ( ω ) | ≤ ϑ −| V (cid:48) | +1 ≤ ϑ −| E (cid:48) | for any ω . When ϑ > 1, use |C ( ω ) | ≤ | V (cid:48) | to see ϑ −| V (cid:48) | + |C ( ω ) | ≤ G ( ω ) = G (cid:48) holds if and only if the Poisson process for eachedge of G (cid:48) contains at least one point. So, (cid:90) [ G ( ω )= G (cid:48) ] d ρ E (cid:48) ,β ( ω ) = (1 − e − β ) | E (cid:48) | , (6.9) and (6.8) follows in the case of cycles.The same bound also holds for the loop model when ϑ |C ( ω ) | is replaced by ϑ |L ( ω ) | in (6.6). For ϑ ≤ ϑ > 1, we usethe inequality |L ( ω ) | ≤ | V (cid:48) | + | ω | that holds for any ω , where | ω | is the number ofbridges in ω . This follows from the fact that each bridge in ω either splits a loopinto two or merges two loops (see Lemma 8.1), and that |L ( ω ) | = | V (cid:48) | when ω = ∅ .Hence, φ ( G (cid:48) ) ≤ (cid:90) ϑ | ω | [ G ( ω )= G (cid:48) ] d ρ E (cid:48) ,β ( ω ) = (cid:16) e − β ∞ (cid:88) n =1 ( ϑβ ) n n ! (cid:17) | E (cid:48) | , (6.10) which gives the bound (6.8) for loops.Combining (6.7) and (6.8) shows that for either loops or cycles, E Λ ,β ( α | V x | ) = (cid:88) C (cid:48) x φ ( C (cid:48) x ) α | V (cid:48) x | (cid:80) G (cid:48) ∩ C (cid:48) x = ∅ φ ( G (cid:48) ) (cid:80) G (cid:48) φ ( G (cid:48) ) ≤ (cid:88) C (cid:48) x α | V (cid:48) x | a | E (cid:48) x | . (6.11) Let δ ( C (cid:48) x ) denote the “depth” of the connected graph C (cid:48) x , i.e., the minimal numberof edges of E (cid:48) x that must be crossed in order to reach any point of V (cid:48) x . Let B ( (cid:96) ) = (cid:88) C (cid:48) x ,δ ( C (cid:48) x ) ≤ (cid:96) α | V (cid:48) x | a | E (cid:48) x | . (6.12) We want an upper bound for B ( (cid:96) ) for any (cid:96) . We show by induction that B ( (cid:96) ) ≤ b for a number b to be determined shortly. We proceed by induction on (cid:96) . The case (cid:96) = 0 is α ≤ b . For (cid:96) + 1, we write the sum over graphs with depth less than (cid:96) + 1, 06 C. GOLDSCHMIDT, D. UELTSCHI, AND P. WINDRIDGE attached at x , as a sum over graphs of depth less than (cid:96) , attached at neighbors of x . Neglecting overlaps gives the following upper bound: B ( (cid:96) + 1) ≤ α (cid:89) y : { x,y }∈E (cid:16) a (cid:88) C (cid:48) y ,δ ( C (cid:48) y ) ≤ (cid:96) α | V (cid:48) y | a | E (cid:48) y | (cid:17) ≤ α (1 + ab ) κ . (6.13) This needs to be less than b ; this condition can be written a ≤ b − (( b/α ) /κ − a is b = α (1 − κ ) − κ . Asufficient condition is then a ≤ ακ (1 − κ ) κ − (6.14) We have obtained that P Λ ,β ( L ( γ x ) > βk ) ≤ α − k +1 (1 − κ ) − κ , (6.15) and this holds for all 1 ≤ α ≤ aκ (1 − κ ) κ − . Choosing the maximal value for α ,we get the bound of the theorem. (cid:3) Suppose T , T , T , . . . areindependent random transpositions of pairs of elements of { , , . . . , n } and π k = T ◦ T ◦ . . . ◦ T k . Write λ ( π k ) for the vector of cycle lengths in π k , sorted intodecreasing order. So, λ i ( π k ) is the size of the i th largest cycle and if there are fewerthan i cycles in π k , we take λ i ( π k ) = 0.Note the simple connection between cycles here and the cycles in our model; if N is a Poisson random variable with mean βn ( n − / 2, independent of the T i , then λ ( π N ) has exactly the distribution of the ordered cycle lengths in C under ρ K n ,β ,where K n is the complete graph with n vertices.Schramm proved that for c > / 2, an asymptotic fraction η ∞ = η ∞ (2 c ) ofelements from { , , . . . , n } lie in infinite cycles of π (cid:98) cn (cid:99) as n → ∞ . The (non-random) fraction η ∞ (2 c ) turns out to be the asymptotic fraction of vertices lyingin the giant component of the Erd˝os-R´enyi random graph with edge probability c/n . Equivalently, η ∞ ( s ) is the survival probability for a Galton-Watson branchingprocess with Poisson offspring distribution with mean s . Berestycki [ ] proved asimilar result.Furthermore, Schramm also showed that the normalised cycle lengths convergeto the Poisson-Dirichlet(1) distribution. Theorem 6.2 (Schramm [ ]) . Let c > / . The law of λ ( π (cid:98) cn (cid:99) ) / ( nη ∞ (2 c )) converges weakly to PD as n → ∞ . 7. Uniform split-merge and its invariant measures We now take a break from spin systems and consider a random evolution onpartitions of [0 , 1] in which blocks successively split or merge. Stochastic processesincorporating the phenomena of coalescence and fragmentation have been muchstudied in the recent probability literature (see, for example, [ 2, 8 ] or Chapter5 of [ ], and their bibliographies). The space of partitions of [0 , 1] provides anatural setting for such processes. The particular model we will discuss here hasthe property that the splitting and merging can be seen to balance each other outin the long run, so that there exists a stationary (or invariant) distribution. Ouraim is to summarise what is known about this invariant distribution. Only a basicfamiliarity with probability theory is assumed and we will recall the essentials as EISENBERG MODELS AND THEIR PROBABILISTIC REPRESENTATIONS 207 we go. This section is self-contained and can be read independently of the first.As is the way among probabilists, we assume there is a phantom probability space(Ω , F , P ) that hosts all our random variables. It is summoned only when needed. Let ∆ denote the space of (decreasing, countable) par-titions of [0 , := (cid:110) p ∈ [0 , N : p ≥ p ≥ . . . , (cid:88) i p i = 1 (cid:111) , (7.1) where the size of the i th part (or block) of p ∈ ∆ is p i . We define split and mergeoperators S ui , M ij : ∆ → ∆ , u ∈ (0 , 1) as follows: • S ui p is the non-increasing sequence obtained by splitting p i into two newparts of size up i and (1 − u ) p i , and • M ij p is the non-increasing sequence obtained by merging p i and p j into apart of size p i + p j . u u M , S u S u M , Figure 6. Illustration for the split-merge process. The partitionundergoes a merge followed by two splits and another merge.The basic uniform split-merge transformation of a partition p is defined asfollows. First we choose two parts of p at random, with the i th part being chosenwith probability p i (this is called size-biased sampling). The two parts, which wecall p I and p J , are chosen independently and we allow repetitions. If the same partis chosen twice, i.e. I = J , sample a uniform random variable U on [0 , 1] and split p I into two new parts of size U p I and (1 − U ) p J (i.e. apply S UI ). If different partsare chosen, i.e. I (cid:54) = J , then merge them by applying M IJ . This transformationgives a new (random) element of ∆ . Conditional on plugging a state p ∈ ∆ intothe transformation, the distribution of the new element of ∆ obtained is given by 08 C. GOLDSCHMIDT, D. UELTSCHI, AND P. WINDRIDGE the so-called transition kernel K ( p, · ) := (cid:88) i p i (cid:90) δ S ui p ( · ) du + (cid:88) i (cid:54) = j p i p j δ M ij p ( · ) . (7.2) Repeatedly applying the transformation gives a sequence P = ( P k ) k =0 , , ,... of random partitions evolving in discrete time. We assume that the updates ateach step are independent. So, given P k , the distribution of P k +1 is independentof P k − , . . . , P . In other words, P is a discrete time Markov process on ∆ withtransition kernel K . We call it the basic split-merge chain.Several authors have studied the large time behaviour of P , and the relatedissue of invariant probability measures, i.e. µ such that µK = µ (if the initial value P is distributed according to µ , then P k also has distribution given by µ at allsubsequent times k = 1 , , . . . ).Recent activity began with Tsilevich [ ]. In that paper the author showed thatthe Poisson-Dirichlet( θ ) distribution (defined in § θ ) with parameter θ = 1 is invariant. The paper contains the conjecture (ofVershik) that PD is the only invariant measure.Uniqueness within a certain class of analytic measures was established byMayer-Wolf, Zerner and Zeitouni in [ ]. In fact they extended the basic split-merge transform described above to allow proposed splits and merges to be re-jected with a certain probability. In particular, splits and merges are proposed asabove but only accepted with probability β s ∈ (0 , 1] and β m ∈ (0 , 1] respectively,independently at different times. The corresponding kernel is K β s ,β m ( p, · ) := β s (cid:88) i p i (cid:90) δ S ui p ( · ) du + β m (cid:88) i (cid:54) = j p i p j δ M ij p ( · )+ (cid:16) − β s (cid:88) i p i − β m (cid:88) i (cid:54) = j p i p j (cid:17) δ p ( · ) . (7.3) We call this ( β s , β m ) split-merge (the basic chain, of course, corresponds to β s = β m = 1). The Poisson-Dirichlet distribution is still invariant, but the parameteris now θ = β s /β m (note that, in fact, any invariant distribution for the chain candepend on β s and β m only through θ since multiplying both acceptance probabilitiesby the same positive constant only affects the speed of the chain).Tsilevich [ ] provided another insight into the large time behaviour of the thebasic split-merge process ( β s = β m = 1). The main theorem is that if P =(1 , , , . . . ) ∈ ∆ , then the law of P , sampled at a random Binomial( n, / n → ∞ .Pitman [ ] studied a related split-merge transformation, and by developingresults of Gnedin and Kerov, reproved Poisson-Dirichlet invariance and refined theuniqueness result of [ ]. In particular, the Poisson-Dirichlet distribution is the onlyinvariant measure under which Pitman’s split-merge transformation composed with‘size-biased permutation’ is invariant.Uniqueness for the basic chain’s invariant measure was finally established byDiaconis, Mayer-Wolf, Zerner and Zeitouni in [ ]. They coupled the split-mergeprocess to a discrete analogue on integer partitions of { , , . . . , n } and then usedrepresentation theory to show the discrete chain is close to equilibrium before de-coupling occurs. EISENBERG MODELS AND THEIR PROBABILISTIC REPRESENTATIONS 209 Schramm [ ] used a different coupling to give another uniqueness proof forthe basic chain. His arguments readily extend to allow β s /β m ∈ (0 , Theorem 7.1. (a) Poisson-Dirichlet( β s /β m ) is invariant for the uniform split-merge chainwith β s , β m ∈ (0 , . (b) If β s /β m ≤ , it is the unique invariant measure. We give a short proof of part (a) in Section 7.3 below. Write M (∆ ) for the set ofprobability measures on ∆ . The Poisson-Dirichlet distribution PD θ ∈ M (∆ ), θ > 0, is a one parameter family of laws introduced by Kingman in [ ]. It hascropped up in combinatorics, population genetics, number theory, Bayesian statis-tics and probability theory. The interested reader may consult [ 20, 34, 5, 43 ] fordetails of applications and extensions. We will simply define it and give some basicproperties.There are two important characterizations of PD θ . We will introduce both,since one will serve to provide intuition and the other will be useful for calcula-tions. We start with the so-called ‘stick-breaking’ construction. Let T , T , . . . beindependent Beta(1 , θ ) random variables (that is, P ( T i > s ) = (1 − s ) θ ; if U is uni-form on [0 , − U /θ is Beta(1 , θ ) distributed). Form a randompartition from the T i by letting the k th block take fraction T k of the unallocatedmass. That is, the first block has size P = T , the second P = T (1 − P ) and P k +1 = T k +1 (1 − P − . . . − P k ). One imagines taking a stick of unit length and break-ing off a fraction T k +1 of what remains after k pieces have already been taken. Aone-line induction argument shows that 1 − P − . . . − P k = (1 − T )(1 − T ) . . . (1 − T k ),giving P k +1 = T k +1 (1 − T )(1 − T ) . . . (1 − T k ) . (7.4) In case it is unclear that (cid:80) ∞ i =1 P i = 1 almost surely, note that E (cid:104) − k (cid:88) i =1 P i (cid:105) = E (cid:104) k (cid:89) i =1 (1 − T i ) (cid:105) = (cid:16)(cid:90) θt (1 − t ) θ − (cid:17) k = ( θ + 1) − k → (7.5) as k → ∞ . So, the vector ( P [1] , P [2] , . . . ) of the P i sorted into decreasing order is anelement of ∆ . It determines a unique measure PD θ ∈ M (∆ ). It is interestingto note that the original vector ( P , P , . . . ) is obtained from ( P [1] , P [2] , . . . ) by size-biased re-ordering; its distribution is called the GEM (Griffiths-Engen-McCloskey)distribution. In other words, consider the interval [0 , 1] partitioned into lengths( P [1] , P [2] , . . . ). Take a sequence U , U , . . . of i.i.d. uniform random variables on[0 , P , P , . . . ).7.2.1. Poisson Point processes. Kingman’s original characterization of PD θ wasmade in terms of a suitable random point process on R + , which is a generalizationof the usual Poisson counting process. We now provide a crash course in the theoryof such processes on a measurable space ( X, B ). (The standard reference is [ ].)Although we will only need this theory for X = R + , there is no extra cost forintroducing it in general. Let M ( X ) denote the set of σ -finite measures on X .Suppose that µ ∈ M ( X ) and consider the special case µ ( X ) < ∞ . Thus, µ ( · ) /µ ( X ) is a probability measure and we can sample, independently, points 10 C. GOLDSCHMIDT, D. UELTSCHI, AND P. WINDRIDGE Y , Y , . . . according to this distribution. Let N be Poisson( µ ( X )) distributed,so that P ( N = n ) = µ ( X ) n n ! e − µ ( X ) . Conceptually, the Poisson point process withintensity measure µ is simply the random collection { Y , . . . , Y N } .Formally, the point process is defined in terms of a random counting measure N which counts the number of random points lying in sets A ∈ B i.e. N ( A ) = (cid:80) N i =1 Y i ∈ A . Thus N ( A ) is a random variable, which has Poisson( µ ( A )) distribu-tion. Indeed, P ( N ( A ) = k ) = ∞ (cid:88) n = k P ( N = n ) P (cid:16) N (cid:88) i =1 Y i ∈ A = k (cid:12)(cid:12)(cid:12) N = n (cid:17) = ∞ (cid:88) n = k µ ( X ) n n ! e − µ ( X ) (cid:18) n ! k !( n − k )! (cid:19) (cid:18) µ ( A ) µ ( X ) (cid:19) k (cid:18) − µ ( A ) µ ( X ) (cid:19) n − k = e − µ ( X ) µ ( A ) k k ! ∞ (cid:88) n = k n − k )! ( µ ( X ) − µ ( A )) n − k = µ ( A ) k k ! e − µ ( A ) . (7.6) Similar calculations show that if A , . . . , A k ∈ B are disjoint then N ( A ) , . . . , N ( A k )are independent. These properties turn out to be sufficient to completely specifythe distribution of the random measure N . Definition 7.1 (Poisson point process) . A Poisson point process on X with inten-sity µ ∈ M ( X ) (or PPP( µ ) for short) is a random counting measure N : B ( X ) → N ∪ { } ∪ {∞} such that • for any A ∈ B ( X ) , N ( A ) has Poisson( µ ( A ) ) distribution. By convention, N ( A ) = ∞ a.s. if µ ( A ) = ∞ . • If A , A , . . . , A k ∈ B are disjoint, the random variables N ( A ) , . . . , N ( A k ) are independent. For general σ -finite intensity measures, we can construct N by superposition.Suppose that X = (cid:83) i X i where the X i are disjoint and µ ( X i ) < ∞ . Use the recipegiven at the start of this section to construct, independently, a PPP( µ | X i ) N i oneach subspace X i . Then N ( A ) = (cid:80) ∞ i =1 N i ( A ) is the desired measure. It is purelyatomic, and the atoms Y , Y , . . . are called the points of the process. In applicationsit is useful to know moments and Laplace transforms of functionals of the process. Lemma 7.2. (1) First moment: If f ≥ or f ∈ L ( µ ) then E (cid:104)(cid:88) i f ( Y i ) (cid:105) = (cid:90) X f ( y ) µ (d y ) (we agree that both sides can be ∞ ). (2) Campbell’s formula: If f ≥ or − e − f ∈ L ( µ ) then E (cid:104) exp (cid:16) − (cid:88) i f ( Y i ) (cid:17)(cid:105) = exp (cid:16) − (cid:90) X (1 − e − f ( y ) ) µ (d y ) (cid:17) (we agree that exp( −∞ ) = 0 ). EISENBERG MODELS AND THEIR PROBABILISTIC REPRESENTATIONS 211 (3) Palm’s formula: Let ˜ M ( X ) ⊂ M ( X ) denote the space of point measureson X ; let G : X × ˜ M → R + be a measurable functional of the points; andsuppose f is as in (2). Then E (cid:104)(cid:88) i f ( Y i ) G ( Y i , N ) (cid:105) = (cid:90) X E [ G ( y, δ y + N )] f ( y ) µ (d y ) . The formulation here is that of Lemma 2.3 of [ ]. We include sketch proofs togive a flavor of the calculations involved. Proof. Let f = (cid:80) nk =1 c k A k , be a simple function with µ ( A k ) < ∞ .(1) We have E (cid:104)(cid:88) i f ( Y i ) (cid:105) = E (cid:104) n (cid:88) k =1 c k N ( A k ) (cid:105) = n (cid:88) k =1 c k µ ( A k ) = (cid:90) X f ( y ) µ (d y ) . (7.7) (2) We have E (cid:104) exp (cid:16) − (cid:88) i f ( Y i ) (cid:17)(cid:105) = E (cid:104) e − (cid:80) k c k N ( A k ) (cid:105) = n (cid:89) k =1 E (cid:104) e − (cid:80) k c k N ( A k ) (cid:105) = n (cid:89) k =1 exp( − µ ( A k )(1 − e − c k )) = exp (cid:16) − (cid:90) X (1 − e − f ( y ) ) µ (d y ) (cid:17) . (7.8) Both (1) and (2) extend to measurable f ≥ f ∈ L ( µ ) follows immediately. Part (2) for 1 − e − f ∈ L ( µ )is also omitted.(3) First suppose G is of the form G ( N ) = exp( − (cid:80) i g ( Y i )) for some non-negative measurable g . Campbell’s formula gives, for q ≥ E (cid:104) exp (cid:16) − q (cid:88) i f ( Y i ) (cid:17) G ( N ) (cid:105) = exp (cid:16) − (cid:90) X (1 − e − qf ( y ) − g ( y ) ) µ (d y ) (cid:17) . (7.9) Differentiating this identity in q at 0 gives E (cid:104)(cid:88) i f ( Y i ) G ( N ) (cid:105) = (cid:90) X f ( y ) e − g ( y ) µ (d y ) exp (cid:16) − (cid:90) X (1 − e − g ( y ) ) µ (d y ) (cid:17) = (cid:90) X f ( y ) e − g ( y ) µ (d y ) E (cid:104) exp (cid:16) − (cid:88) i g ( Y i ) (cid:17)(cid:105) = (cid:90) X f ( y ) E (cid:104) exp (cid:16) − (cid:88) i g ( Y i ) − g ( y ) (cid:17)(cid:105) µ (d y )= (cid:90) X f ( y ) E [ G ( N + δ y )] µ (d y ) , (7.10) where Campbell’s formula is used to get the second and last lines.Now, suppose G ( y, N ) = (cid:80) nk =1 c k y ∈ A k exp( − (cid:80) i g k ( Y i )) for A , . . . , A n ∈ B and measurable g k : X → [0 , ∞ ). By linearity, the preceding calculations give 12 C. GOLDSCHMIDT, D. UELTSCHI, AND P. WINDRIDGE E (cid:104)(cid:88) i f ( Y i ) G ( Y i , N ) (cid:105) = (cid:90) X n (cid:88) k =1 c k y ∈ A k f ( y ) E (cid:104) exp (cid:16) − (cid:88) i g k ( Y i ) − g k ( y ) (cid:17)(cid:105) µ (d y )= (cid:90) X f ( y ) E [ G ( y, N + δ y )] µ (d y ) . (7.11) From here it is a standard monotone class argument. (cid:3) The Poisson-Dirichlet distribution via a PPP. Consider the PPP withintensity measure given by η (d x ) = θx − exp( − x )d x on [0 , ∞ ). (Note that η isan infinite measure, but is σ -finite since η (2 − k − , − k ] ≤ θ .) A practical way toconstruct this process is given in Tavar´e [ ]. Let T < T < . . . be the points of aPoisson counting process of rate θ (that is, the differences T i +1 − T i are independentexponential variables of rate θ ) and E , E , . . . be exponentially distributed withrate 1. Then, the points in our PPP( η ) can be expressed as ξ i = exp( − T i ) E i , i ≥ xη ( x ) |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| ||| | || |||||| ||| | | | | | | Figure 7. A sample of the Poisson Point Process (points markedby bars) with intensity measure η (overlaid in gray). Note that thepoints are dense around the origin.The probability that all points are less than K > P ( N ( K, ∞ ) = 0) = exp (cid:16) − (cid:90) ∞ K θx − exp( − x )d x (cid:17) → (7.12) as K → ∞ . Thus, there is a largest point and we can order the points in decreasingorder so that ξ ≥ ξ ≥ . . . ≥ 0. The sum (cid:80) ∞ i =1 ξ i is finite almost surely. Indeed,we can say much more. Recall that the Gamma( γ, λ ) distribution has density1Γ( γ ) λ γ x γ − exp( − λx ) . Lemma 7.3. We have ∞ (cid:88) i =1 ξ i ∼ Gamma( θ, . EISENBERG MODELS AND THEIR PROBABILISTIC REPRESENTATIONS 213 Proof. Since (cid:80) i ξ i is a non-negative random variable, its distribution is de-termined by its Laplace transform. By Campbell’s formula, this is given by E (cid:104) exp (cid:16) − r (cid:88) i ξ i (cid:17)(cid:105) = exp (cid:16) − θ (cid:90) ∞ (1 − e − rx ) x − exp( − x )d x (cid:17) = exp (cid:16) − θ (cid:90) r (cid:90) ∞ exp( − x (1 + r ))d x d r (cid:17) = (1 + r ) − θ , (7.13) for | r | < 1, implying that (cid:80) i ξ i is Gamma( θ, 1) distributed. (cid:3) The Poisson-Dirichlet( θ ) distribution, PD θ ∈ M (∆ ), is the law of the orderedpoints, normalised by their sum, i.e.1 (cid:80) i ξ i ( ξ , ξ , ξ , . . . ) . (7.14) In the next section, we will wish to appeal to various properties of Beta andGamma random variables which are often known collectively as the “Beta-Gammaalgebra”. Recall that the Beta( a, b ) distribution has density Γ( a + b )Γ( a )Γ( b ) t a (1 − t ) b on[0 , Lemma 7.4. Suppose that Γ λα ∼ Gamma( α, λ ) and Γ λβ ∼ Gamma( β, λ ) are inde-pendent. Then • Γ λα + Γ λβ ∼ Gamma( α + β, λ ) , • Γ λα / (Γ λα + Γ λβ ) ∼ Beta( α, β ) , • The two random variables above are independent. Note that the converse also follows: if B ∼ Beta( α, β ) is independent of Γ λα + β ∼ Gamma( α + β, λ ) then B Γ λα + β ∼ Gamma( α, λ ), (1 − B )Γ λα + β ∼ Gamma( β, λ ) andthese last two random variables are independent. Proof. In order to simplify the notation, let X = Γ λα and Y = Γ λβ . We willfind the joint density of S = X + Y and R = X/ ( X + Y ). We first find the Jacobiancorresponding to this change of variables: we have ∂x∂s = r, ∂x∂r = s∂y∂s = 1 − r, ∂y∂r = − s and so the Jacobian is |− rs − (1 − r ) s | = s . Noting that X = RS and Y = (1 − R ) S ,we see that S and R have joint density s α ) λ α ( rs ) α − e − λrs β ) ((1 − r ) s ) β − e − λ (1 − r ) s = 1Γ( α + β ) λ α + β s α + β − e − λs · Γ( α + β )Γ( α )Γ( β ) r α − (1 − r ) β − . (7.15) Since this factorizes with the factors being the correct Gamma and Beta densities,the result follows. (cid:3) 14 C. GOLDSCHMIDT, D. UELTSCHI, AND P. WINDRIDGE In the next lemma, we will see the power of the Beta-Gamma algebra. We useit to make a connection between our two different representations of the Poisson-Dirichlet distribution. This will serve as a warm up for the calculations in the nextsection. Lemma 7.5. Suppose that P = ( P , P , . . . ) ∼ PD θ . Let P ∗ be a size-biased pickfrom amongst P , P , . . . . Then P ∗ ∼ Beta(1 , θ ) . So P ∗ has the same distribution as the length of the first stick in the stick-breaking construction. Proof. Note that, conditional on P , P , . . . , we have that P ∗ = P i with probability P i , i ≥ . (7.16) In order to determine the distribution of P ∗ , it suffices to find E [ f ( P ∗ )] for allbounded measurable test functions f : [0 , → R + . (Indeed, it would suffice to find E [ f ( P ∗ )] for all functions of the form f ( x ) = exp( − qx ) i.e. the Laplace transform.However, our slightly unusual formulation will generalize better when we considerrandom variables on ∆ in the next section.) Conditioning on P , P , . . . and usingthe Tower Law we see that E [ f ( P ∗ )] = E [ E [ f ( P ∗ ) | P , P , . . . ]] = E (cid:104) ∞ (cid:88) i =1 P i f ( P i ) (cid:105) . (7.17) Now use the representation (7.14) to see that this is equal to E (cid:104) ∞ (cid:88) i =1 ξ i (cid:80) ∞ j =1 ξ j f (cid:16) ξ i (cid:80) ∞ k =1 ξ k (cid:17)(cid:105) . (7.18) This is in a form to which we can apply the Palm formula; we obtain E (cid:104)(cid:90) ∞ yy + (cid:80) ∞ i =1 ξ i f (cid:16) yy + (cid:80) ∞ j =1 ξ j (cid:17) θy − e − y d y (cid:105) . (7.19) After cancelling y and y − , we recognise the density of the Exp(1) (= Gamma(1,1))distribution and so we can write E (cid:104) θ Γ + (cid:80) ∞ i =1 ξ i f (cid:16) ΓΓ + (cid:80) ∞ j =1 ξ j (cid:17)(cid:105) , (7.20) where Γ ∼ Exp(1) is independent of ξ , ξ , . . . . Recall that (cid:80) ∞ i =1 ξ i ∼ Gamma( θ, (cid:80) ∞ i =1 ξ i has a Gamma( θ + 1 , 1) distribution and is inde-pendent of Γ / (Γ + (cid:80) ∞ i =1 ξ i ), which has a Beta(1 , θ ) distribution. Hence, we get E (cid:104) θ Γ + (cid:80) ∞ i =1 ξ i (cid:105) E (cid:2) f ( B ) (cid:3) , (7.21) where B ∼ Beta(1 , θ ). We conclude by observing that E (cid:104) θ Γ + (cid:80) ∞ i =1 ξ i (cid:105) = 1 . (7.22) (cid:3) We close this section by noting an important property of the PPP we use tocreate the Poisson-Dirichlet vector. EISENBERG MODELS AND THEIR PROBABILISTIC REPRESENTATIONS 215 Lemma 7.6. The random variable (cid:80) ∞ i =1 ξ i is independent of (cid:80) i ξ i ( ξ , ξ , ξ , . . . ) . This is another manifestation of the independence in the Beta-Gamma algebra;see [ ]. We use the methodthat we exploited in the proof of Lemma 7.5 to prove part (a) of Theorem 7.1.First define a random function F : ∆ → ∆ corresponding to ( β s , β m ) split-merge as follows. Fix p ∈ ∆ and let I ( p ) and J ( p ) be the indices of the twoindependently size-biased parts of p , that is P ( I ( p ) = k ) = P ( J ( p ) = k ) = p k , k ≥ . (7.23) Now let U and V be independent U(0 , 1) random variables, independent of I ( p )and J ( p ). Let F ( p ) = S Ui p if I ( p ) = J ( p ) = i and V ≤ β s M ij p if I ( p ) = i (cid:54) = J ( p ) = j and V ≤ β m p otherwise . (7.24) We wish to prove that if P ∼ PD θ then F ( P ) ∼ PD θ also. Let g : ∆ → R + bea bounded measurable test function which is symmetric in its arguments (this justmeans that we can forget about ordering the elements of our sequences). Then,conditioning on P , considering the different cases and using the Tower Law, wehave E [ g ( F ( P ))] = E (cid:104) E (cid:104) V ≤ β s ∞ (cid:88) i =1 I ( P )= J ( P )= i g ( S Ui P ) (cid:12)(cid:12)(cid:12) P (cid:105)(cid:105) + E (cid:104) E (cid:104) V >β s ∞ (cid:88) i =1 I ( P )= J ( P )= i g ( P ) (cid:12)(cid:12)(cid:12) P (cid:105)(cid:105) + E (cid:104) E (cid:104) V ≤ β m (cid:88) i (cid:54) = j I ( P )= i J ( P )= j g ( M ij P ) (cid:12)(cid:12)(cid:12) P (cid:105)(cid:105) + E (cid:104) E (cid:104) V >β m (cid:88) i (cid:54) = j I ( P )= i J ( P )= j g ( P ) (cid:12)(cid:12)(cid:12) P (cid:105)(cid:105) . (7.25) Note that, conditional on P , I ( P ) = i, J ( P ) = j with probability P i P j , so that weget E [ g ( F ( P ))] = β s E (cid:104) ∞ (cid:88) i =1 P i g ( S Ui P ) (cid:105) + (1 − β s ) E (cid:104) ∞ (cid:88) i =1 P i g ( P ) (cid:105) + β m E (cid:104)(cid:88) i (cid:54) = j P i P j g ( M ij P ) (cid:105) + (1 − β m ) E (cid:104)(cid:88) i (cid:54) = j P i P j g ( P ) (cid:105) . (7.26) Now use the symmetry of g to write g ( S Uk P ) = g (( P k U, P k (1 − U ) , ( P i ) i ≥ ,i (cid:54) = k )) (7.27) and g ( M ij P ) = g (( P i + P j , ( P k ) k ≥ ,k (cid:54) = i,j )) . (7.28) 16 C. GOLDSCHMIDT, D. UELTSCHI, AND P. WINDRIDGE Set ( P , P , . . . ) = (cid:80) ∞ i =1 ξ i ( ξ , ξ , . . . ) as in (7.14) to obtain E [ g ( F ( P ))] = β s E (cid:20) ∞ (cid:88) k =1 ξ k (cid:0)(cid:80) ∞ i =1 ξ i (cid:1) g (cid:16) (cid:80) ∞ i =1 ξ i ( ξ k U, ξ k (1 − U ) , ( ξ i ) i ≥ ,i (cid:54) = k ) (cid:17)(cid:21) + (1 − β s ) E (cid:20) ∞ (cid:88) k =1 ξ k ( (cid:80) ∞ i =1 ξ i ) g (cid:16) (cid:80) ∞ i =1 ξ i ( ξ i ) i ≥ (cid:17)(cid:21) + β m E (cid:20)(cid:88) i (cid:54) = j ξ i ξ j ( (cid:80) ∞ k =1 ξ k ) g (cid:16) (cid:80) ∞ k =1 ξ k ( ξ i + ξ j , ( ξ k ) k ≥ ,k (cid:54) = i,j ) (cid:17)(cid:21) + (1 − β m ) E (cid:20) (cid:88) i (cid:54) = j ξ i ξ j ( (cid:80) ∞ k =1 ξ k ) g (cid:16) (cid:80) ∞ i =1 ξ i ( ξ i ) i ≥ (cid:17)(cid:21) . (7.29) The Palm formula (Lemma 7.2, (3)) applied to each of the expectations above(twice for the double sums) gives E [ g ( F ( P ))] = θβ s E (cid:20)(cid:90) ∞ x − e − x x ( x + (cid:80) ∞ k =1 ξ k ) g (cid:16) x + (cid:80) ∞ k =1 ξ k ( xU, x (1 − U ) , ( ξ i ) i ≥ ) (cid:17) d x (cid:21) + θ (1 − β s ) E (cid:20)(cid:90) ∞ x − e − x x ( x + (cid:80) ∞ k =1 ξ k ) g (cid:16) x + (cid:80) ∞ k =1 ξ k ( x, ( ξ i ) i ≥ ) (cid:17) d x (cid:21) + θ β m E (cid:20)(cid:90) ∞ (cid:90) ∞ x − e − x y − e − y xy ( x + y + (cid:80) ∞ k =1 ξ k ) g (cid:16) x + y + (cid:80) ∞ k =1 ξ k ( x + y, ( ξ i ) i ≥ ) (cid:17) d x d y (cid:21) + θ (1 − β m ) E (cid:20)(cid:90) ∞ (cid:90) ∞ x − e − x y − e − y xy ( x + y + (cid:80) ∞ k =1 ξ k ) g (cid:16) x + y + (cid:80) ∞ k =1 ξ k ( x, y, ( ξ i ) i ≥ ) (cid:17) d x d y (cid:21) . (7.30) It helps to recognise the densities we are integrating over here (after cancellation).In the first two expectations, which correspond to split proposals, we have the den-sity xe − x of the Gamma(2,1) distribution. The other density to appear is e − x e − y ,which corresponds to a pair of independent standard exponential variables. UsingLemma 7.4, it follows that E [ g ( F ( P ))] = θβ s E (cid:20) (cid:80) ∞ k =1 ξ k ) g (cid:16) 1Γ + (cid:80) ∞ k =1 ξ k (Γ U, Γ(1 − U ) , ( ξ i ) i ≥ ) (cid:17)(cid:21) + θ (1 − β s ) E (cid:20) (cid:80) ∞ k =1 ξ k ) g (cid:16) 1Γ + (cid:80) ∞ k =1 ξ k (Γ , ( ξ i ) i ≥ ) (cid:17)(cid:21) + θ β m E (cid:20) (cid:80) ∞ k =1 ξ k ) g (cid:16) 1Γ + (cid:80) ∞ k =1 ξ k (Γ , ( ξ i ) i ≥ ) (cid:17)(cid:21) + θ (1 − β m ) E (cid:20) (cid:80) ∞ k =1 ξ k ) g (cid:16) 1Γ + (cid:80) ∞ k =1 ξ k (Γ U, Γ(1 − U ) , ( ξ i ) i ≥ ) (cid:17)(cid:21) , (7.31) where Γ ∼ Gamma(2 , ξ i ) i ≥ . By Lemmas 7.4 and 7.6, Γ + (cid:80) k ξ k is Gamma(2 + θ, 1) distributed and independent of the argument of g in allof the above expectations. More calculation shows that E (cid:20) (cid:80) ∞ k =1 ξ k ) (cid:21) = 1 θ ( θ + 1) , (7.32) EISENBERG MODELS AND THEIR PROBABILISTIC REPRESENTATIONS 217 and so we are left with E [ g ( F ( P ))] = θβ s + θ (1 − β m ) θ ( θ + 1) E (cid:20) g (cid:16) 1Γ + (cid:80) ∞ k =1 ξ k (Γ U, Γ(1 − U ) , ( ξ i ) i ≥ ) (cid:17)(cid:21) + θ (1 − β s ) + θ β m θ ( θ + 1) E (cid:20) g (cid:16) 1Γ + (cid:80) ∞ k =1 ξ k (Γ , ( ξ i ) i ≥ ) (cid:17)(cid:21) . (7.33) Next use β s = θβ m to get θβ s + θ (1 − β m ) = θ and θ (1 − β s ) + θ β m = θ. (7.34) So the expression for E [ g ( F ( P ))] simplifies to θ ( θ + 1) E (cid:20) g (cid:16) 1Γ + (cid:80) ∞ k =1 ξ k (Γ U, Γ(1 − U ) , ( ξ i ) i ≥ ) (cid:17)(cid:21) + 1( θ + 1) E (cid:20) g (cid:16) 1Γ + (cid:80) ∞ k =1 ξ k (Γ , ( ξ i ) i ≥ ) (cid:17)(cid:21) . (7.35) We can re-express this as a sum of expectations as follows:1 θ ( θ + 1) E (cid:20)(cid:90) ∞ (cid:90) ∞ θ e − x e − y g (cid:16) x + y + (cid:80) ∞ k =1 ξ k ( x, y, ( ξ i ) i ≥ ) (cid:17) d x d y (cid:21) + 1 θ ( θ + 1) E (cid:20)(cid:90) ∞ θxe − x g (cid:16) x + (cid:80) ∞ k =1 ξ k ( x, ( ξ i ) i ≥ ) (cid:17) d x d y (cid:21) . (7.36) Using the Palm formula in the other direction gives1 θ ( θ + 1) E (cid:20)(cid:88) i (cid:54) = j ξ i ξ j g (cid:16) (cid:80) ∞ k =1 ξ k ( ξ k ) k ≥ (cid:17) + ∞ (cid:88) k =1 ξ k g (cid:16) (cid:80) ∞ k =1 ξ k ( ξ k ) k ≥ (cid:17)(cid:21) = 1 θ ( θ + 1) E (cid:20)(cid:16) ∞ (cid:88) k =1 ξ k (cid:17) g (cid:16) (cid:80) ∞ k =1 ξ k ( ξ k ) k ≥ (cid:17)(cid:21) . (7.37) Once again, (cid:80) ∞ k =1 ξ k is independent of the argument of g . Moreover, it is easilyshown that E (cid:20)(cid:16) ∞ (cid:88) k =1 ξ k (cid:17) (cid:21) = θ ( θ + 1) , (7.38) since it is simply the second moment of a Gamma( θ, 1) random variable. Thus, E [ g ( F ( P ))] = E [ g ( P )] , (7.39) from which the result follows. The dynamics in the next sectionwill be in continuous time, so we close this section by describing a continuous timeversion of the split-merge process. First, consider the standard Poisson countingprocess ( N t , t ≥ { , , , . . . } , are piecewise constant, increasing and rightcontinuous. At each integer k , it is held for an exponentially distributed randomtime before jumping to k + 1. Consequently, only finitely many jumps are madeduring each finite time interval. We say N t increments at rate 1.Continuous time split-merge is the process ( P N t , t ≥ 0) obtained by composing( P k , k = 0 , , , , . . . ) with an independent Poisson counting process. It is aMarkov process in ∆ with the following dynamics. Suppose the present state is p ∈ ∆ . Attach to each part p i an exponential alarm clock of rate β s p i and to each 18 C. GOLDSCHMIDT, D. UELTSCHI, AND P. WINDRIDGE pair ( p i , p j ) of distinct parts a clock of rate 2 β m p i p j . Wait for the first clock toring. If p i ’s clock rings first then split p i uniformly (i.e. apply S Ui with U uniform).If the alarm for ( p i , p j ) rings first then apply M ij . In other words, part p i splitsuniformly at rate β s p i and distinct parts p i and p j merge at rate 2 β m p i p j . Due tothe memoryless property of the exponential distribution, once an alarm clock hasrung, all of the alarm clocks are effectively reset, and the process starts over fromthe new state.More formally, define the rate kernel Q : ∆ × B (∆ ) → [0 , ∞ ) by Q ( p, · ) := β s (cid:88) i p i (cid:90) δ S ui p ( · ) du + β m (cid:88) i (cid:54) = j p i p j δ M ij p ( · ) (7.40) and the (uniformly bounded) ‘rate of leaving’ q : ∆ → [0 , ∞ ) q ( p ) := Q ( p, ∆ ) = β s (cid:88) i p i + β m (cid:88) i (cid:54) = j p i p j . (7.41) Using standard theory (e.g. Proposition 12.20, [ ]), there exists a Markov processon ∆ that waits for an Exponential( q ( p )) amount of time in state p before jumpingto a new state chosen according to Q ( p, · ) /q ( p ). Furthermore, since K β s ,β m ( p, · ) = Q ( p, · ) + (1 − q ( p )) δ p ( · ) , (7.42) this process is constructed explicitly as ( P N t , t ≥ Lemma 7.7. A measure ν ∈ M (∆ ) is invariant for the continuous time process ( P N t , t ≥ if, and only if, it is invariant for ( P k , k = 0 , , , , . . . ) . 8. Effective split-merge process of cycles and loops This section contains an heuristic argument that connects the loop and cyclemodels of section 6.1 and the split-merge process in section 7.4. The heuristicleads to the conjecture that the asymptotic normalized lengths of the cycles andloops have Poisson-Dirichlet distribution. By looking at the rates of the effectivesplit-merge process, we can identify the parameter of the distribution.Consider the cycle or loop model on the cubic lattice Λ n = { , . . . , n } d in Z d .As hinted at in section 6.1, we expect that macroscopic cycles emerge for inversetemperatures β large enough as n → ∞ . Of course, we believe this also holds for anysequence of sufficiently connected graphs (Λ n ) with diverging number of vertices,but for simplicity we restrict attention to cubic lattices. Furthermore, since thesame arguments apply to both the cycle and loop models, we focus on cycles andonly mention the modifications for loops when necessary.Denote by λ ( i ) the length of the i th longest cycle, and recall that η macro ( β ) isthe fraction of sites lying in macroscopic cycles (see Section 4.4). Conjecture 8.1. Suppose d ≥ . There exists β c > such that for β > β c : (a) The fractions of sites in infinite and macroscopic cycles (or loops) ap-proach the same typical value, and η := η ∞ ( β ) = η macro ( β ) > . EISENBERG MODELS AND THEIR PROBABILISTIC REPRESENTATIONS 219 (b) The vector of ordered normalised cycle lengths (cid:18) λ (1) η n d , λ (2) η n d , . . . (cid:19) converges weakly to a random variable ξ in ∆ as n → ∞ . Assuming the conjectured result is true, what is the distribution of ξ ? In somerelated models (the random-cluster model), ξ has been found to be the trivial (andnon-random!) partition (1 , , , . . . ). However, we conjecture that there are many macroscopic cycles in our model (rather than a unique giant cycle) and that theirrelative lengths can be described explicitly by the Poisson-Dirichlet distribution. Conjecture 8.2. The distribution of ξ in Conjecture 8.1 (b) is PD θ for an appro-priate choice of θ . The rest of this section is concerned with justifying this conjecture. The readermay guess what the parameter θ should be. We will tease it out below and identifyit in section 8.4.See Section 6.3 for a summary of rigorous results by Schramm to support thisconjecture on the complete graph. Recall that P Λ n ,β,ϑ denotes the prob-ability measure for either the loop or cycle model. We define an ergodic Markovprocess on Ω with P Λ ,β,ϑ as invariant measure. The process evolves by adding orremoving bridges to the current configuration. Conveniently, the effect of such anoperation is to either split a cycle or merge two cycles. Lemma 8.1. Suppose ω ∈ Ω and ω (cid:48) is ω with either a bridge added (i.e. ω (cid:48) = ω ∪ { ( e, t ) } for some ( e, t ) ∈ E × [0 , β ] ) or a bridge removed (i.e. ω (cid:48) = ω − { ( e, t ) } for some ( e, t ) ∈ ω ). Then C ( ω (cid:48) ) is obtained by splitting a cycle or merging twocycles in C ( ω ) . Similarly, L ( ω (cid:48) ) is obtained by a split or merge in L ( ω ) . The point is that adding or removing a bridge never causes, for example, severalcycles to join, a cycle to split into many pieces or the cycle structure to remainunchanged. removingaddingbridgebridgeremoving addingbridgebridge Figure 8. Adding or removing bridges always split or merge cy-cles. Up to topological equivalence, this figure lists all possibilities.The Lemma is most easily justified by drawing pictures for the different cases.Suppose that we add a new bridge. Either both endpoints of the new bridge belongto the same cycle or two different cycles. In the former case, the cycle is split andwe say the bridge is a self-contact. In the latter case, the two cycles are joined andthe bridge is called a contact between the two cycles. This is illustrated in Figure8 for cycles and Figure 9 for loops. 20 C. GOLDSCHMIDT, D. UELTSCHI, AND P. WINDRIDGE Suppose that we remove an existing bridge. Again, either both of the bridge’sendpoints belong to the same cycle (self-contact) or they are in different cycles(contact between the two cycles). In the former case, removal splits the cycle andin the latter, the two cycles are joined.As this argument hints, it is helpful to formally define the ‘contacts’ betweencycles. Suppose that γ ∈ C ( ω ) is a cycle. Recall from Section 3.1 that this means γ ( τ ) = ( x ( τ ) , t ( τ )), τ ≥ V × [0 , β ] per , where x is piecewiseconstant and has a jump discontinuity across the edge e = ( x ( τ − ) , x ( τ )) ∈ E attime τ if, and only if, the bridge ( e, t ( τ )) is present in ω . Such bridges are called selfcontact bridges, the set of which is denoted B γ . Removing a bridge from B γ ⊂ ω causes γ to split. removing addingbridgebridgeremovingaddingbridgebridge Figure 9. Same as Figure 8, but for loops instead of cycles.The self contact zone C γ of γ is the set of ( e, τ ) ∈ E × [0 , β ] for which e =( x ( τ ) , x ( τ + jβ )) for some integer j , i.e. the ( e, t ) bridge touches different legs of γ ’s trajectory and so adding a bridge from C γ splits γ .The contact bridges B γ,γ (cid:48) and zones C γ,γ (cid:48) between distinct cycles γ, γ (cid:48) ∈ C ( ω )are defined similarly. Specifically, B γ,γ (cid:48) ⊂ ω is comprised of bridges in ω thatare traversed by γ = ( x, t ) and γ (cid:48) = ( x (cid:48) , t (cid:48) ), i.e. ( e, t ) ∈ ω such that e = ( x ( t + j β ) , x (cid:48) ( t + j β )) for some integers j , j . Removal of a bridge in B γ,γ (cid:48) causes γ and γ (cid:48) to merge. C γ,γ (cid:48) is the set of ( e, t ) ∈ E × [0 , β ] such that e = ( x ( t + j β ) , x ( t + j β )) forsome j , j , i.e. those bridges that would merge γ and γ (cid:48) . Note that the contact(and self contact) zones partition E × [0 , β ] while the contact bridges partition ω . The promised P Λ ,β,ϑ -invariant Markov process, denoted( X t ) t ≥ is defined as follows. Suppose that α > • A new bridge appears in ( e, dt ) at rate ϑ α dt if its appearance causes acycle to split and at rate ϑ − α dt if it causes two cycles to join. • An existing bridge is removed at rate ϑ − α if its removal causes a cycleto split and at rate ϑ − (1 − α ) if its removal causes two cycles to join. • No other transitions occur.The rates are not uniformly bounded, so a little effort is required to check X is well behaved (does not ‘explode’). Accepting this, we can show X is actuallyreversible with respect to our cycle model. Lemma 8.2. The unique invariant measure of X is P Λ ,β,ϑ . The proof is straightforward and so we omit it.In the sequel we take α = 1 / 2, so that adding and removing bridges occur atthe same rates. EISENBERG MODELS AND THEIR PROBABILISTIC REPRESENTATIONS 221 As weknow, adding or removing bridges causes cycles to split or merge so the dynamics( C ( X t ) , t ≥ 0) that X induces on cycles is a kind of coagulation-fragmentation pro-cess. However, these dynamics are not Markovian and depend on the underlyingprocess in a complicated manner. Ideally we would like a simpler, more transparentdescription for the dynamics. The first step towards this is to rewrite the transitionrates for X in terms of the contact zones and bridges.Suppose that X is currently in state ω ∈ Ω. A cycle γ ∈ C ( ω ) splits if either abridge from C γ is added, or a bridge from B γ ⊂ ω is removed. The total rate atwhich these transitions occur is √ ϑ ( | B γ | + | C γ | ) , (8.1) where | C γ | = (cid:80) e ∈E Leb( { t ∈ [0 , β ] : ( e, t ) ∈ C γ } ) is the (one-dimensional) Lebesguemeasure of the self contact zone. Two distinct cycles γ and γ (cid:48) merge if a bridgefrom C γ,γ (cid:48) is added or one from B γ,γ (cid:48) removed. The combined rate is √ ϑ − ( | B γ,γ (cid:48) | + | C γ,γ (cid:48) | ) , (8.2) where | C γ,γ (cid:48) | = (cid:80) e ∈E Leb( { t ∈ [0 , β ] : ( e, t ) ∈ C γ,γ (cid:48) } ).8.3.1. Heuristics. We believe that, for suitably connected graphs and largeenough β , cycles should be macroscopic. The trajectories of these cycles shouldspread evenly over all edges and vertices in the graph. In particular, macroscopiccycles should come into contact with each other many times and we expect someaveraging phenomenon to come into play. The longer a cycle is, on average, themore intersections with other cycles it should have. In particular, we believe thecontact zone between two macroscopic cycles should have size proportional to thecycles’ length.That is, if γ and γ (cid:48) are cycles with lengths λ and λ (cid:48) respectively then there isa ‘law of large numbers’ | C γ | ∼ c λ , | B γ | ∼ c λ (8.3) and | C γ,γ (cid:48) | ∼ c λλ (cid:48) , | B γ,γ (cid:48) | ∼ c λλ (cid:48) , (8.4) for constants c and c (the notation X ∼ Y means that the ratio of the randomvariables converges to 1 in probability as Λ n grows).The constants may depend on ϑ and β and the graph geometry. We believethey are linear in β but do not depend on ϑ . Note that the size of the contact zonescan be calculated easily for the complete graph. We get | C γ,γ (cid:48) | = βλλ (cid:48) , | C γ | = β λ ( λ − . (8.5) In the case ϑ = 1, we also have numerical support for | B γ,γ (cid:48) | ∼ βλλ (cid:48) , | B γ | ∼ β λ . (8.6) Continuing with the heuristic, C ( X ) is ‘nearly’ a Markov process in which cycles split and merge. Substituting(8.3) into (8.1) and (8.4) into (8.2), and multiplying by 2 √ ϑ ( c + c ) (which justchanges the speed of the process, not its invariant measure) we see that a cycle oflength λ splits at rate ϑλ , while two cycles with lengths λ and λ (cid:48) merge at rate2 λλ (cid:48) . There seems no reason to suppose that splits are not uniform. 22 C. GOLDSCHMIDT, D. UELTSCHI, AND P. WINDRIDGE Suddenly there are many similarities between C ( X ) and the continuous timesplit-merge process of section 7.4. This suggests that Poisson-Dirichlet PD θ islurking somewhere in the normalised cycle length distribution. What is the rightchoice of the parameter θ ?Write ϑ = β s /β m , β s , β m ∈ (0 , 1] and multiply the rates by β m to see thata cycle of length λ splits uniformly at rate β s λ , while two cycles with lengths λ and λ (cid:48) merge at rate 2 β m λλ (cid:48) . Up to the normalising factor (which is close to theconstant η macro | Λ n | ), these are exactly the rates in section 7.4. Thus, the parameter θ should be equal to ϑ . This fact was initially not obvious. References [1] M. Aizenman and B. Nachtergaele. Geometric aspects of quantum spin states. Comm. Math.Phys. , 164(1):17–63, 1994.[2] D. Aldous. Deterministic and stochastic models for coalescence (aggregation and coagulation):a review of the mean-field theory for probabilists. Bernoulli , 5(1):3–48, 1999.[3] G. Alon and G. Kozma. The probability of long cycles in interchange processes.http://arxiv.org/abs/1009.3723, 2010.[4] O. Angel. Random infinite permutations and the cyclic time random walk. In Discrete randomwalks (Paris, 2003) , Discrete Math. Theor. Comput. Sci. Proc., AC, pages 9–16 (electronic).Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2003.[5] R. Arratia, A. D. Barbour, and S. Tavar´e. Logarithmic combinatorial structures: a proba-bilistic approach . EMS Monographs in Mathematics. European Mathematical Society (EMS),Z¨urich, 2003.[6] N. Berestycki. Emergence of giant cycles and slowdown transition in random transpositionsand k -cycles. Electr. J. Probab. , 16:152–173, 2011.[7] V. L. Berezinskiˇı. Destruction of long-range order in one-dimensional and two-dimensionalsystems having a continuous symmetry group i. classical systems. Soviet J. Exper. Theor.Phys. , 32:493–500, 1971.[8] J. Bertoin. Random fragmentation and coagulation processes , volume 102 of Cambridge Stud-ies in Advanced Mathematics . Cambridge University Press, Cambridge, 2006.[9] V. Betz and D. Ueltschi. Spatial random permutations and Poisson-Dirichlet law of cyclelengths. Electr. J. Probab. , 16:1173–1192, 2011.[10] M. Biskup. Reflection positivity and phase transitions in lattice spin models. In Methods ofcontemporary mathematical statistical physics , volume 1970 of Lecture Notes in Math. , pages1–86. Springer, Berlin, 2009.[11] C. Borgs, R. Koteck´y, and D. Ueltschi. Low temperature phase diagrams for quantum per-turbations of classical spin systems. Comm. Math. Phys. , 181(2):409–446, 1996.[12] L. Chayes, L. P. Pryadko, and K. Shtengel. Intersecting loop models on Z d : rigorous results. Nuclear Phys. B , 570(3):590–614, 2000.[13] J. G. Conlon and J. P. Solovej. Upper bound on the free energy of the spin 1 / Lett. Math. Phys. , 23(3):223–231, 1991.[14] N. Crawford and D. Ioffe. Random current representation for transverse field Ising model. Comm. Math. Phys. , 296(2):447–474, 2010.[15] N. Datta, R. Fern´andez, and J. Fr¨ohlich. Low-temperature phase diagrams of quantum latticesystems. I. Stability for quantum perturbations of classical systems with finitely-many groundstates. J. Statist. Phys. , 84(3-4):455–534, 1996.[16] P. Diaconis, E. Mayer-Wolf, O. Zeitouni, and M. P. W. Zerner. The Poisson-Dirichlet lawis the unique invariant distribution for uniform split-merge transformations. Ann. Probab. ,32(1B):915–938, 2004.[17] F. J. Dyson, E. H. Lieb, and B. Simon. Phase transitions in quantum spin systems withisotropic and nonisotropic interactions. J. Statist. Phys. , 18(4):335–383, 1978.[18] P. Erd˝os and A. R´enyi. On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutat´oInt. K¨ozl. , 5:17–61, 1960.[19] W. G. Faris. Outline of quantum mechanics. In Entropy and the quantum , volume 529 of Contemp. Math. , pages 1–52. Amer. Math. Soc., Providence, RI, 2010. EISENBERG MODELS AND THEIR PROBABILISTIC REPRESENTATIONS 223 [20] S. Feng. The Poisson-Dirichlet distribution and related topics . Probability and its Applica-tions (New York). Springer, Heidelberg, 2010. Models and asymptotic behaviors.[21] G. B. Folland. Real analysis . Pure and Applied Mathematics (New York). John Wiley &Sons Inc., New York, second edition, 1999. Modern techniques and their applications, AWiley-Interscience Publication.[22] J. Fr¨ohlich, R. Israel, E. H. Lieb, and B. Simon. Phase transitions and reflection positivity.I. General theory and long range lattice models. Comm. Math. Phys. , 62(1):1–34, 1978.[23] J. Fr¨ohlich, R. B. Israel, E. H. Lieb, and B. Simon. Phase transitions and reflection positivity.II. Lattice systems with short-range and Coulomb interactions. J. Statist. Phys. , 22(3):297–347, 1980.[24] J. Fr¨ohlich, B. Simon, and T. Spencer. Infrared bounds, phase transitions and continuoussymmetry breaking. Comm. Math. Phys. , 50(1):79–95, 1976.[25] J. Fr¨ohlich and T. Spencer. The Kosterlitz-Thouless transition in two-dimensional abelianspin systems and the Coulomb gas. Comm. Math. Phys. , 81(4):527–602, 1981.[26] D. Gandolfo, J. Ruiz, and D. Ueltschi. On a model of random cycles. J. Stat. Phys. ,129(4):663–676, 2007.[27] J. Ginibre. Existence of phase transitions for quantum lattice systems. Comm. Math. Phys. ,14(3):205–234, 1969.[28] G. R. Grimmett. Space-time percolation. In In and out of equilibrium. 2 , volume 60 of Progr.Probab. , pages 305–320. Birkh¨auser, Basel, 2008.[29] T. E. Harris. Nearest-neighbor Markov interaction processes on multidimensional lattices. Advances in Math. , 9:66–89, 1972.[30] D. Ioffe. Stochastic geometry of classical and quantum Ising models. In Methods of contempo-rary mathematical statistical physics , volume 1970 of Lecture Notes in Math. , pages 87–127.Springer, Berlin, 2009.[31] O. Kallenberg. Foundations of modern probability . Probability and its Applications (NewYork). Springer-Verlag, New York, second edition, 2002.[32] T. Kennedy, E. H. Lieb, and B. S. Shastry. Existence of N´eel order in some spin- Heisenbergantiferromagnets. J. Statist. Phys. , 53(5-6):1019–1030, 1988.[33] J. F. C. Kingman. Random discrete distributions. J. Roy. Statist. Soc. Ser. B , 37:1–15, 1975.With a discussion by S. J. Taylor, A. G. Hawkes, A. M. Walker, D. R. Cox, A. F. M. Smith,B. M. Hill, P. J. Burville, T. Leonard and a reply by the author.[34] J. F. C. Kingman. Mathematics of genetic diversity , volume 34 of CBMS-NSF RegionalConference Series in Applied Mathematics . Society for Industrial and Applied Mathematics(SIAM), Philadelphia, Pa., 1980.[35] J. F. C. Kingman. Poisson processes , volume 3 of Oxford Studies in Probability . The Claren-don Press Oxford University Press, New York, 1993. Oxford Science Publications.[36] J. M. Kosterlitz and D. J. Thouless. Ordering, metastability and phase transitions in two-dimensional systems. Journal of Physics C: Solid State Physics , 6(7):1181, 1973.[37] E. Mayer-Wolf, O. Zeitouni, and M. P. W. Zerner. Asymptotics of certain coagulation-fragmentation processes and invariant Poisson-Dirichlet measures. Electron. J. Probab. , 7:no.8, 25 pp. (electronic), 2002.[38] N. D. Mermin and H. Wagner. Absence of ferromagnetism or antiferromagnetism in one- ortwo-dimensional isotropic heisenberg models. Phys. Rev. Lett. , 17(22):1133–1136, Nov 1966.[39] B. Nachtergaele. Quantum spin systems after DLS 1978. In Spectral theory and mathematicalphysics: a Festschrift in honor of Barry Simon’s 60th birthday , volume 76 of Proc. Sympos.Pure Math. , pages 47–68. Amer. Math. Soc., Providence, RI, 2007.[40] E. J. Neves and J. F. Perez. Long range order in the ground state of two-dimensional anti-ferromagnets. Physics Letters A , 114(6):331 – 333, 1986.[41] J. Pitman. Poisson-Dirichlet and GEM invariant distributions for split-and-merge transfor-mation of an interval partition. Combin. Probab. Comput. , 11(5):501–514, 2002.[42] J. Pitman. Combinatorial stochastic processes , volume 1875 of Lecture Notes in Mathematics .Springer-Verlag, Berlin, 2006. Lectures from the 32nd Summer School on Probability Theoryheld in Saint-Flour, July 7–24, 2002, With a foreword by Jean Picard.[43] J. Pitman and M. Yor. The two-parameter Poisson-Dirichlet distribution derived from astable subordinator. Ann. Probab. , 25(2):855–900, 1997.[44] D. Ruelle. Statistical mechanics: Rigorous results . W. A. Benjamin, Inc., New York-Amsterdam, 1969. 24 C. GOLDSCHMIDT, D. UELTSCHI, AND P. WINDRIDGE [45] O. Schramm. Compositions of random transpositions. Israel J. Math. , 147:221–243, 2005.[46] B. Simon. The statistical mechanics of lattice gases. Vol. I . Princeton Series in Physics.Princeton University Press, Princeton, NJ, 1993.[47] S. Tavar´e. The birth process with immigration, and the genealogical structure of large pop-ulations. J. Math. Biol. , 25(2):161–168, 1987.[48] B. T´oth. Improved lower bound on the thermodynamic pressure of the spin 1 / Lett. Math. Phys. ∼ balint/oktatas/statisztikus fizika/jegyzet/, 1996.[50] N. Tsilevich. On the simplest split-merge operator on the infinite-dimensional simplex. Arxivpreprint math/0106005 , 2001.[51] N. V. Tsilevich. Stationary random partitions of a natural series. Teor. Veroyatnost. i Prime-nen. , 44(1):55–73, 1999. 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